Tig.  1 


INTRODUCTION 


TO 


ASTRONOMY 


DESIGNED  AS  A 


TEXT-BO 


FOB  THE  USE  OF 


STUDENTS   IN   COLLEGE. 


BY 

DENISON   OLMSTED,  LL.D., 

LATE   PROFESSOR   OF   NATURAL   PHILOSOPHY  AND   ASTRONOMY   IN   YALE   COLLEGE. 


THIRD     STEREOTYPE     EDITION. 
REVISED    BY 

E.  S.  SNELL,  LL.D., 

PROFESSOR   OF   NATURAL   PHILOSOPHY   IN   AMHERST   COLLEGE. 


NEW  YORK : 
COLLINS  &  BKOTHEB, 

106    LEONARD    STREET. 
1866. 


Entered  according  to  Act  of  Congress,  in  the  year  1S44, 

BY  DENISON  OLMSTED, 
In  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


REVISED  EDITION. 
Entered  according  to  Act  of  Congress,  in  the  year  1861, 

BY  JULIA  M.  OLMSTED, 

FOR  THE  CHILDREN  OF  DENISON  OLMSTED,  DECEASED, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Connecticut 


THIRD  STEREOTYPE  EDITION. 
Entered  according  to  Act  of  Congress,  in  the  year  1866, 

BY  JULIA  M.  OLMSTED, 

FOR  THE  CHILDREN  OF  DENISON  OLMSTED,  DECEASED, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Connecticut 


JOHN    G.  SHEA, 

Stereotyper  and  Electrotyper, 

New  York. 

ALVORD,  Printer. 


AUTHOR'S  PREFACE. 


NEARLY  all  who  have  written  Treatises  on  Astronomy,  designed  for 
young  learners,  appear  to  have  erred  in  one  of  two  ways  :  thev  have 
either  disregarded  demonstrative  evidence,  and  relied  on  mere  popular 
illustration,  or  they  have  exhibited  the  elements  of  the  science  innaked 
mathematical  formula.  The  former  are  usually  diffuse  and  superficial ; 
the  latter,  technical  and  abstruse. 

In  the  following  Treatise,  we  have  endeavored  to  unite  the  advantages 
of  both  methods.  We  have  sought,  first,  to  establish  the  great  princi- 
ples of  astronomy  on  a  mathematical  basis ;  and,  secondly,  to  render  the 
study  interesting  and  intelligible  to  the  learner,  by  easy  and  familiar 
illustrations.  We  would  not  encourage  any  one  to  believe  that  he  can 
enjoy  a  full  view  of  the  grand  edifice  of  astronomy,  while  its  noble 
foundations  are  hidden  from  his  sight ;  nor  would  we  assure  him  that 
he  can  contemplate  the  structure  in  its  true  magnificence,  while  its 
basement  alone  is  within  his  field  of  vision.  We  would,  therefore,  that 
the  student  of  astronomy  should  confine  his  attention  neither  to  the 
exterior  of  the  building,  nor  to  the  mere  analytic  investigation  of  its 
structure.  We  would  desire  that  he  should  not  only  study  it  in  models 
and  diagrams,  and  mathematical  formula,  but  should  at  the  same  time 
acquire  a  love  of  nature  herself,  and  cultivate  the  habit  of  raising  his 
views  to  the  grand  originals.  Nor  is  the  effort  to  form  a  clear  concep- 
tion of  the  motions  and  dimensions  of  the  heavenly  bodies  less  favorable 
to  the  improvement  of  the  intellectual  powers  than  the  study  of  pure 
geometry. 

But  it  is  evidently  possible  to  follow  out  all  the  intricacies  of  an  ana- 
lytical process,  and  to  arrive  at  a  full  conviction  of  the  great  truths  of 
astronomy,  and  yet  know  very  little  of  nature.  According  to  our  ex- 
perience, however,  but  few  students  in  the  course  of  a  liberal  education 
will  feel  satisfied  with  this.  They  do  not  need  so  much  to  be  convinced 
that  the  assertions  of  astronomers  are  true,  as  they  desire  to  know  what 
the  truths  are, 'and  how  they  were  ascertained;  and  they  will  derive 
from  the  study  of  astronomy  little  of  that  moral  and  intellectual  eleva- 
tion which  they  had  anticipated,  unless  they  learn  to  look  upon  the 
heavens  with  new  views,  and  a  clear  comprehension  of  their  wonderful 
mechanism. 

Much  of  the  difficulty  that  usually  attends  the  early  progress  of  the 


IV  PKEFACE. 

astronomical  student  arises  from  his  being  too  soon  introduced  to  the 
most  perplexing  part  of  the  whole  subject — the  planetary  motions.  In 
this  work,  the  consideration  of  these  is  for  the  most  part  postponed  until 
the  learner  has  become  familiar  with  the  artificial  circles  of  the  sphere, 
and  conversant  with  the  celestial  bodies.  We  then  first  take  the  most 
simple  view  possible  of  the  planetary  motions  by  contemplating  them  as 
they  really  are  in  nature,  and  afterward  proceed  to  the  more  difficult 
inquiry,  why  they  appear  as  they  do.  Probably  no  science  derives  such 
signal  advantage  from  a  happy  arrangement  as  astronomy  ; — an  order 
wkich  brings  out  every  fact  or  doctrine  of  the  science  just  in  the  place 
where  the  mind  of  the  learner  is  prepared  to  receive  it. 


i 


PREFACE  TO  THE  REVISED  EDITION  OF  1866. 


THE  revision  of  Professor  Olmstead's  Astronomy  made  in  1861, 
owing  to  circumstances  which  I  could  not  well  control,  was  not  so 
thorough  as  I  desired.  The  present  edition  is  more  in  accordance  with 
the  idea  which  was  in  my  mind,  but  which  was  then  only  partially 
carried  out.  The  changes  now  introduced  are  principally  of  the  same 
nature  as  those  made  in  the  revision  of  the  College  Natural  Philosophy. 
Some  historical  matter  is  omitted  ;  a  few  alterations  are  made  in  the 
arrangement  of  topics ;  the  subject  of  nearly  every  paragraph  is  pre- 
sented in  an  italic  heading ;  and,  throughout,  I  have  aimed  to  state 
principles  and  describe  phenomena  with  more  conciseness,  where  I 
thought  I  could  do  it  with  equal  clearness.  About  one-half  of  the  en- 
gravings are  new.  A  part  of  these  are  for  the  illustration  of  points 
either  not  illustrated  or  not  discussed  before,  and  a  part  are  substituted 
for  those  which  were  faulty.  The  true  proportions  of  lines  and  angles 
can  not,  in  general,  be  presented  in  astronomical  diagrams  ;  and  the  pupil 
must  depend  mainly  on  his  teacher  for  the  help  which  he  needs  in  this 
respect.  In  a  very  few  cases,  where  it  seemed  practicable,  a  figure  is 
added,  simply  for  the  purpose  of  exhibiting  the  true  size  of  an  angle, 
and  the  correct  proportion  of  lines.  A  few  pages  at  the  end  of  the 
volume  are  occupied  with  tables,  which  may  be  found  useful. 

I  am  much  indebted  to  Professor  Henry  H.  White,  of  the  Kentucky 
University,  who  has  very  kindly  aided  me  by  numerous  criticisms  and 
suggestions,  of  which  I  have  made  free  use  in  this  revision. 

E.  S.  SNELL. 
AMHERST  COLLEGE,  March,  1866. 


CONTENTS. 


CHAPTER  I. 

FA«B 

Astronomy. — Its  subject. — Globular  form  of  the  earth  proved. — Modes  of 
measuring  the  earth. — The  terrestrial  equator. — The  horizon  and  seconda- 
ries.— The  celestial  equator. — The  ecliptic. — The  diurnal  motion. — Its  phe- 
nomena.— Problems  on  the  globes 1-14 

CHAPTER  II. 

Parallax.— Diurnal  parallax.— Its  variation.— To  find  the  parallax  of  the  moon. 
— Atmospheric  refraction. — Illumination  of  the  sky. — Twilight 15-23 

CHAPTER  III. 

The  observatory. — The  transit-instrument. — The  astronomical  clock. — Measur- 
ing right  ascension. — The  mural  circle. — Measuring  declination. — Altitude 
and  azimuth  instruments.— The  sextant. — Spherical  problems 24-38 

CHAPTER  IV. 

Observations  of  the  sun's  place.— The  ecliptic  and  zodiac. — The  annual  mo- 
tion.— The  change  of  seasons. — Arrangement  of  heat  and  cold. — Form  of  the 
earth's  orbit. — Mode  of  determining  it 38-47 

CHAPTER  V. 

The  sidereal  and  solar  day. — Mean  and  apparent  solar  time. — Reasons  why 
solar  days  are  unequal.— The  equation  of  time. — The  calendar 47-54 

CHAPTER  VI. 

Projectile,  centripetal,  and  centrifugal  forces. — Laws  of  centrifugal  force. — Its 
effects  on  the  earth. — Loss  of  weight. — Spheroidal  form. — Proofs  of  diurnal 
motion 54-62 

CHAPTER  VII. 

The  sun. — Its  form. — Its  distance. — Its  dimensions. — Its  rotation. — Solar  spots. 
— Theory  of  spots.— Condition  of  the  sun's  surface. — The  zodiacal  light 62-69 


Vlll  CONTENTS. 

CHAPTER  VIII. 

PAG  a 

Kepler's  laws. — Law  of  areas  proved. — Law  of  gravity  proved. — Its  prevalence 
throughout  the  system. — The  paths  of  projectiles. — Effect  of  an  impulse  on 
one  body  of  a  system 69-80 

CHAPTER  IX. 

Precession  of  equinoxes. — Consequent  motion  of  the  poles. — Cause. — Compo- 
sition of  rotations. — The  tropical  and  sidereal  years. — Nutation. — Aber- 
ration of  light. — Velocity  of  light  discovered. — Advance  of  apsides. — Its 
cause. — How  to  find  the  sun's  true  place 80-88 

CHAPTER  X. 

The  moon. — Its  distance  and  size. — Its  motion  round  the  earth. — Its  orbit. — 
Librations. — Its  path  about  the  sun. — Its  phases. — The  harvest  moon. — 
The  moon's  surface. — Measurement  of  its  mountains. — Appearance  of  the 
earth  from  the  moon 88-102 

CHAPTER  XI. 

The  moon's  motion  disturbed  by  the  sun. — Gravity  to  the  earth  diminished. 
— Equations  for  finding  the  moon's  place. — Equation  of  the  center. — Evec- 
tion. — Variation. — Annual  equation. — Advance  of  apsides. — Ketrogradation 
of  nodes. — Periodical  and  secular  equations 102-109 

CHAPTER  XII. 

Eclipses. — Their  cause. — Eclipse  months. — The  earth's  shadow. — Its  dimen- 
sions computed. — To  find  beginning,  middle,  and  end  of  a  lunar  eclipse. — 
Eclipse  of  the  sun. — Dimensions  of  the  moon's  shadow. — Its  velocity  over 
the  earth. — The  Saros.— Phenomena  of  a  solar  eclipse 109-124 

CHAPTER  XIII. 

Methods  of  determining  longitude. — By  the  chronometer. — By  eclipses. — By 
the  lunar  method. — By  the  telegraph. — Change  of  days  in  going  round  the 
earth 125-129 

CHAPTER  XIV. 

Tides. — Form  of  equilibrium  under  the  action  of  the  moon.— Joint  action  of 
sun  and  moon.— Diurnal  inequalities.— Effect  of  coasts.— Tides  in  seas  and 
lakes 190-135 

CHAPTER  XV. 

Planets  grouped. — Distances  from  the  sun.— Revolutions. — Dimensions. — 
Masses  and  densities. — Mercury. — Its  motions. — Its  phases. — Its  transits. — 
Venus.— Its  transits.— Parallax  of  the  sun  found.— Mars.— Its  motions. . . .  135-151 


CONTENTS.  IX 

CHAPTER  XVI. 

PACK 

The  planetoids. — Jupiter. — Its  belts. — Its  satellites. — Their  eclipses  and  oc- 
cultations. — The  velocity  of  light  found  by  them. — Saturn. — Its  rings. — 
Their  disappearances. — The  satellites  of  Saturn. — Uranus. — Its  satellites. 
Neptune. — Its  discovery 152-165 

CHAPTER  XVII. 

Elements  of  a  planetary  orbit. — Method  of  finding  the  first. — The  second. — 
The  third.— The  fourth.— The  fifth  and  sixth.— The  masses  of  the  planets 
found. — Perturbations. — In  the  positions  of  orbits. — In  their  forms. — Sta- 
bility of  the  system.— Relations  of  the  planets 165-180 

CHAPTER  XVIII. 

Comets. — Their  number. — Effects  of  eccentricity  of  orbit, — Dimensions  of 
comets. — Their  masses. — How  to  find  their  orbits. — Halley's  comet. — 
Comets  of  short  period. — A  resisting  medium. — Remarkable  comets. — 
Shooting  stars. — Meteoric  showers. — Aerolites 180-194 

CHAPTER  XIX. 

The  stellar  universe. — Classifications  of  stars. — Constellations. — Annual  par- 
allax.— Stars  whose  distance  is  known. — Nature  of  fixed  stars. — Proper 
motions. — Double  stars. — Binary  stars. — Their  orbits. — Their  masses. — 
Periodic  stars. — Clusters. — Nebulae. — The  galaxy. — The  Nebular  hypothe 
sis  . .  . .  194-213 


ASTRONOMICAL  TABLES 214-218 

1* 


ASTRONOMY. 


CHAPTER  I. 

GENEEAL   FOEM   AND   DIMENSIONS   OF  THE   EAETH. — THE 
DIUENAL   MOTION. — AETIFICIAL  GLOBES. 

1.  General  definitions. — Astronomy  is  the   science  which 
treats  of  the  heavenly  bodies — that  is,  of  the  sun,  the  planets 
and  their  satellites,  the  comets,  and  the  fixed  stars. 

The  sun,  planets,  satellites,  and  comets  constitute  the  solar 
system,  which  is  so  called  because  the  sun  is  the  principal  body 
belonging  to  it,  and  controls  the  movements  of  all  the  others. 

The  fixed  stars  are  the  bodies  situated  at  vast  distances  out- 
side of  the  solar  system,  and  which,  on  account  of  that  distance, 
exhibit  little  or  no  change  of  position  with  respect  to  each 
other. 

2.  The  Copernican  system. — This  name  is  given,  in  honor  of 
Copernicus,  to  the  science  of  astronomy  as  now  established  by 
demonstration,  in  distinction  from  the  erroneous  systems  of  the 
ancients.     It  explains  the  diurnal  and  annual  motions  of  the 
heavens,  by  supposing  the  earth  to  rotate  each  day  on  its  axis, 
and  to  revolve  once  a  year  around  the  sun. 

3.  The  globular  form  of  the  earth. — That  the  earth  is  nearly, 
if  not  exactly  a  sphere,  is  indicated  in  several  ways. 

1.  It  is  one  of  the  planets.  And,  as  we-see  the  other  planets 
to  be  nearly  spherical,  we  reason  from  analogy  that  the  earth  is 
spherical  also. 

1 


DIP   OF   THE   HORIZON. 


2.  In  a  lunar  eclipse,  whichever  side  is  turned  toward  the 
moon,  the  outline  of  its  shadow,  projected  on  that  body,  is 
always  circular. 

3.  Its  convexity,  by  which  it  wholly  or  partially  conceals 
distant  objects,  as  a  lighthouse  or  a  ship  at  sea,  appears  to  be 
equally  great  on  all  parts  of  the  ocean. 

4.  An  arc  of  a  given  number  of  miles,  measured  on  any  part 
of  the  earth,  is  found  always  to  subtend  an  angle  of  nearly 
equal  size  at  the  center ;  showing  that  the  curvature  is  every- 
where nearly  the  same. 

5.  The  depression,  or  dip  of  the  horizon,  is  equally  great  at 
every  place,  and  on  every  side  of  the  observer,  provided  his 
elevation  above  the  ocean  level  is  the  same.     This  will  be  un- 
derstood by  the  next  article. 


Fig.  1. 


4.  Dip  of  the  horizon.  — If  the  eye  were  at  A  (Fig.  1)  on 
the  surface  of  the  earth,  the  vault  of  the  heavens  would  be  lim- 
ited by  a  plane  touching  the  earth  at 
A,  and  would  therefore  be  just  a  hemi- 
sphere. But  if  the  eye  is  elevated,  as 
to  O,  and  tangent  lines  are  drawn  from 
that  point  to  the  earth  on  every  side, 
then  more  than  a  hemisphere  of  the 
sky  is  visible.  Let  ZC  be  the  direc- 
tion of  a  plumb-line,  and  let  HOR, 
represent  a  plane  perpendicular  to  it ; 
then  there  would  be  a  celestial  hemi- 
sphere in  view  above  this  plane,  and 
the  remotest  visible  points  on  the  earth 
would  be  depressed  below  the  plane  by 
the  angle  HOD  or  KOE.  This  angle 
is  called  the  dip  of  the  horizon.  If  AO  is  a  given  height, 
it  is  found  that  the  angle  HOD  is  sensibly  equal  on  whatever 
side  of  the  station,  or  on  whatever  part  of  the  earth,  the 
measurement  is  made.  It  follows  from  this  that  the  earth  is 
very  nearly  a  sphere. 

At  the  height  of  100  feet,  the  depression  is  about  10',  and 
varies  nearly  as  the  square  root  of  the  height. 

The  word  down  expresses  the  direction  in  which  a  plumb- 


DIMENSIONS  OF  THE   EARTH. 


3 


line  hangs,  or  a  body  falls — that  is,  toward  the  center  of  the 
earth.  Hence,  on  different  parts  of  the  earth,  "  down"  denotes 
all  possible  directions.  So  "  up,"  or  from  the  center,  is  in  every 
direction ;  and  the  direction  which  is  down  at  one  place,  is  up 
at  a  place  on  the  opposite  side  of  the  earth. 

5.  Dimensions  of  the  earth. — The  semi-diameter  of  the 
earth  may  be  approximately  found  by  measuring  the  height  of 
the  station  AO  (Fig.  1),  and  the 
length  of  the  tangent  line  OD.  If  O 
were  the  summit  of  a  mountain,  then 
D  would  be  the  most  distant  point 
from  which  it  could  be  discerned.  In 
Fig.  2,  suppose  that  the  height  of  the 
mountain  BD,  and  the  distance  to  the 
point  where  it  is  just  seen  in  the  hori- 
zon AD,  have  been  measured.  Let 
BD  =  A,  and  AD  =  d,  and  the  radius, 


=  a?a  -f  2  A  a?  +  A2.    Hence,  2  A  x  =  d?  —  A2,  and  x  =      - ,  — . 

zA 

Thus,  the  semi-diameter  of  the  earth  is  found  in  terms  of  A 
and  d. 

The  magnitude  of  the  earth  may  be  more  accurately  found, 
by  measuring  the  arc  of  a  meridian.  Let  a  line  be  carefully 
measured  due  north  on  the  earth's  surface,  and  the  correspond- 
ing difference  of  latitude  be  observed,  as  indicated  by  the 
change  in  the  elevation  of  the  stars.  Then,  the  surveyed  line 
is  the  same  part  of  the  earth's  circumference,  which  the  differ- 
ence of  latitude  is  of  360°.  Thus,  if  the  arc  is  1°  30',  its  length 
is  found  to  be  about  103.5  miles.  Hence, 

1°  30':  360°::  103.5:  24,840; 

which  is  nearly  the  number  of  miles  in  the  circumference  of 
the  earth.  By  a  comparison  of  the  most  accurate  measure- 
ments, it  is  ascertained  that 

The  circumference  of  the  earth        =24,857  miles. 

The  diameter  (24,857  -r-  3.14159+)  =  7,912.4  miles. 

One  degree  of  the  circumference     =  365,000  feet. 

One  second  =  about  100  feet. 


4  SECONDARIES   OF   THE   EQUATOR. 

6.  Inequalities  of  surface. — Although   the   surface  of  the 
earth  is  uneven,  and  there  are  high  mountains  and  deep  valleys 
in  many  parts  of  it,  yet  these  are  very  minute  compared  with 
the  magnitude  of  the  entire  earth ;  so  that  the  spherical  form 
is  not  disturbed  by  their  existence.     Mountains,  four  or  five 
miles  high  on  the  earth,  are  relatively  no  more  than  are  the 
particles  of  dust  which  adhere  to  a  globe  one  foot  in  diameter. 
Thin  writing-paper,  pasted  upon  such  a  globe  in  the  form  of 
the  continents,  would  be  sufficiently  thick  to  represent  their 
general  elevation  above  the  oceans. 

7.  The  diurnal  rotation. — The  earth  revolves  continually 
from  west  to  east,  on  an  imaginary  line  drawn  through  its  cen- 
ter, called  the  earth's  axis.     The  time  occupied  in  completing  a 
revolution  is  called  a  day,  which  is  divided  into  twenty-four 
hours.     A  great  circle  of  the  earth,  perpendicular  to  the  axis, 
is  called  the  equator.     In  the  diurnal  rotation,  every  particle  of 
the  earth  describes  a  circle,  whose  plane  is  either  parallel  to 
the  equator  or  coincident  with  it.     The  extremities  of  the  axis 
are  called  respectively  the  north  and  south  poles. 

8.  Secondaries  of  the  equator. — All  great  circles  passing 
through  the  poles,  and  therefore  perpendicular  to  the  equator, 
are  called  meridians.     Such  a  circle  may  be  supposed  to  pass 
through  any  place  whatever  on  the  earth,  and  is  called  the  me- 
ridian of  that  place.     As  all  great  circles  of  a  sphere  which  are 
perpendicular  to  a  given  great  circle,  are  called  its  secondaries, 
the  meridians  are  secondaries  of  the  equator. 

The  latitude  of  a  place  is  its  distance  north  or  south  from  the 
equator,  measured  on  the  meridian  of  that  place,  in  degrees, 
minutes,  and  seconds.     Parallels  of  latitude  are  small  circles" 
of  the  earth,  parallel  to  the  equator. 

The  longitude  of  a  place  is  the  distance  of  its  meridian  in 
degrees,  minutes,  and  seconds,  east  or  west  from  some  standard 
meridian,  as  that  of  the  observatory  of  Greenwich.  The  people 
of  different  nations  usually  reckon  longitude  from  some  import- 
ant observatory  of  their  own  country.  Thus,  the  French  reckon 
from  Paris,  and  the  Americans  from  Washington.  Any  place 
on  the  earth  is  determined  by  giving  its  latitude  and  longitude. 


THE   HORIZON  AND   ITS  SECONDARIES.  5 

9.  The  celestial  sphere. — The  earth  is  called  the  terrestrial 
sphere.     The  celestial  sphere  is  that  apparent  vault,  called  the 
sky,  which  surrounds  the  earth  on  every  side,  and  to  which  all 
the  heavenly  bodies  seem  to  be  attached.     The  center  of  the 
earth  is  regarded  as  the  center  of  the  celestial  sphere  also.    But 
the  distance  of  nearly  all  the  heavenly  bodies  is  so  immense, 
that  it  is  immaterial  from  what  point  of  the  earth  they  are 
viewed.     Hence,  for  most  purposes  of  astronomy,  the  eye  of 
the  observer  may  be  considered  as  the  center  of  the  celestial 
sphere. 

10.  The  horizon  and  its  secondaries. — If  the   plumb-line 
(usually  called  the  vertical),  at  any  place  on  the  earth,  is  sup- 
posed to  be  extended  till  it  intersects  the  celestial  sphere,  it 
marks  the  zenith  above  the  place,  and  the  nadir  below  it. 
And  a  plane  passed  through  the  center  of  the  earth,  perpendic- 
ular to  the  vertical,  is  called  the  rational  horizon  of  that  place. 
This  is  a  great  circle  of  the  celestial  sphere,  and  divides  it  into 
upper  and  lower  hemispheres.     The  sensible  horizon  is  parallel 
to  the  rational  horizon,  and  passes  through  the  place  on  the 
earth's  surface.     The  planes  of  these  two  horizons  are  therefore 
near  4,000  miles  apart ;  but  so  great  is  the  distance  of  the 
heavenly  bodies,  that  the  two  planes  seem  to  unite  in  the  same 
great  circle  of  the  heavens. 

If  the  observer  is  at  all  elevated  above  the  earth's  surface, 
the  boundary  line  between  sky  and  water  is  a  little  lower  than 
the  horizon,  so  that  somewhat  more  than  half  of  the  celestial 
sphere  is  in  view.  (Art.  4.)  The  secondaries  of  the  horizon 
intersect  each  other  in  the  vertical  line,  and  are  called  vertical 
circles.  One  of  them  is  the  meridian  of  the  place.  The  inter- 
sections of  the  meridian  and  horizon  are  the  north  and  south 
points  of  compass.  The  vertical  circle  at  right  angles  to  the 
meridian  is  called  the  prime  vertical.  This  intersects  the  hori- 
zon in  the  points  called  east  and  west. 

The  altitude  of  a  heavenly  body  is  its\  elevation  above  the 
horizon,  measured  on  the  vertical  circle  passing  through  the 
body.  The  zenith  distance  of  a  body  is  the  distance  between 
it  and  the  zenith,  and  is  therefore  the  complement  of  'its 
altitude. 


6 


CELESTIAL   EQUATOR. 


The  azimuth  of  a  heavenly  body  is  an  arc  of  the  horizon, 
measured  from  the  meridian  to  the  vertical  circle,  which  passes 
through  the  body.  The  amplitude  is  measured  from  the  verti- 
cal circle  passing  through  the  body  to  the  prime  vertical,  and 
is  therefore  the  complement  of  the  azimuth.  The  altitude,  or 
zenith  distance  of  a  heavenly  body,  along  with  its  azimuth  or 
amplitude,  determines  its  place  in  the  visible  heavens. 

1 1 .  The  celestial  equator  and  its  secondaries. — If  the  axis  on 
which  the  earth  revolves  is  produced  to  the  heavens,  it  becomes 
the  axis  of  the  celestial  sphere,  and  marks  the  north  and  south 
poles  of  that  sphere.  The  north  pole  is  at  present  in  the  con- 
stellation of  Ursa  Minor.  If  the  plane  of  the  equator  be  ex- 
tended in  like  manner,  it  becomes  the  celestial  equator.  The 
secondaries  to  this  circle  are  called  meridians,  as  on  the  earth. 
They  are  also  called  hour-circles,  because  the  arcs  of  the 
equator  intercepted  between  them  are  used  as  measures  of 

time. 

Fig.  3. 


ZD 


LR 


Let  n  (Tig.  3)  represent  the  north  pole  of  the  earth,  s  its 
south  pole,  ^the  equator  (projected  in  a  straight  line),  o  a  given 


THE   ECLIPTIC.  7 

place  whose  north  latitude  is  eo.  Then  N,  S,  are  the  poles  of 
the  celestial  sphere,  EQ  is  the  celestial  equator,  Z  is  the  zenith 
of  the  place  o,  B,  is  its  nadir,  and  HO  its  rational  horizon. 
oesqn  is  the  terrestrial  meridian  of  the  same  place,  and 
ZESQN  is  its  celestial  meridian,  or  hour-circle.  * 

1 2.  The  ecliptic. — Besides  the  equator,  there  is  an  import- 
ant circle  of  the  celestial  sphere,  called  the  ecliptic.     It  is  that 
in  which  the  sun  appears  to  make  its  annual  circuit  around  the 
heavens.     It  is  inclined  to  the  equator  at  an  angle  of  nearly 
23£°,  crossing  it  in  two  opposite  points,  called  the  equinoctial 
points,  or  equinoxes.     The  word  "  equinoxes"  is  used  also  to 
express  the  times  at  which  the  sun  crosses  the  equator,  because 
at  those  times  the  nights  are  equal  to  the  days.     The  vernal 
equinox  is  the  time  when  the  sun  passes  the  equator  from  south 
to  north,  as  it  occurs  in  the  spring,  about  March  21st.     The 
autumnal  equinox  occurs  on  or  near  September  22d,  when  the 
sun  returns  to  the  south  of  the  equator. 

The  solstitial  points,  or  solstices,  are  those  points  of  the 
ecliptic,  which  are  furthest  north  or  south  from  the  equator, 
situated  therefore  midway  between  the  equinoxes.  They  are 
so  named,  because  there  the  sun  stops  in  his  advance  north- 
ward or  southward,  and  begins  to  return.  The  summer  solstice 
is  the  point  where,  and  also  the  time  when  the  sun  is  furthest 
north,  about  the  22d  of  June.  He  passes  the  winter  solstice  on 
or  near  the  22d  of  December. 

The  equinoctial  colure  is  that  secondary  to  the  equator 
which  passes  through  the  equinoxes.  The  solstitial  colure  is 
that  which  passes  through  the  solstices.  They  are  therefore  at 
right  angles  to  each  other,  and  the  latter  is  a  secondary  to  the 
ecliptic,  as  well  as  to  the  equator. 

13.  Signs  of  the  ecliptic. — The  ecliptic  is   divided  into 
12  equal  parts  of  30°  each,  called  signs,  which,  beginning  at 
the  vernal  equinox,  succeed  each  other  eastward,  in  the  follow- 
ing order : 


8  DIURNAL   MOTION  OF   THE  HEAVENS. 

Northern.  Southern. 

1.  Aries      .     .  .  v  7.  Libra      ...=== 

2.  Taurus    ...»  8.  Scorpio  .     .  .  TO 

3.  Gemini  .     .  .  n  9.  Sagittarius  .  # 

4.  Cancer    .     .  .  ®  10.  Capri corrnis  .  ^ 

5.  Leo    .     .     .  .  *l  11.  Aquarius     .  .  £? 

6.  Virgo      .     .  .  m  12.  Pisces     .     .  .  * 

The  vernal  equinox  being  at  the  first  point  of  Aries,  the  sum- 
mer solstice  is, at  the  first  of  Cancer,  the  autumnal  equinox  at 
the  first  of  Libra,  and  the  winter  solstice  at  the  first  of  Capricorn. 

1 4.  R'gTit  ascension  and  declination. — The  right   ascen- 
sion of  a  heavenly  body  is  the  angular  distance  of  its  meridian 
from  the  vernal  equinox,  measured  eastward  on  the  equator. 
The  declination  of  a  body  is  its  angular  distance  north  or  south 
from  the  equator,  measured  on  the  meridian  of  the  body. 

The  equator  is  the  plane  of  reference  for  right  ascension  and 
declination  on  the  celestial  sphere,  as  it  is  for  latitude  and 
longitude  on  the  terrestrial.  But  terrestrial  longitude  is  reck- 
oned both  east  and  west,  while  right  ascension  is  reckoned  only 
to  the  east. 

15.  Celestial    longitude   and   latitude. — On   the   celestial 
sphere,  longitude  and  latitude  are  referred  to  the  ecliptic,  not 
to  the  equator.     Suppose  a  secondary  to  the  ecliptic  to  pass 
through  a  heavenly  body ;  the  distance  of  the  body  from  the 
ecliptic,  measured  on  the  secondary,  is  its  latitude  ;  and  the  dis- 
tance of  this  secondary  from  the  vernal   equinox,  measured 
eastward  on  the  ecliptic,  is  its  longitude. 

Right  ascension  and  longitude  are  reckoned  only  eastward, 
from  0°  to  360°,  the  first  on  the  equator,  the  other  on  the 
ecliptic. 

1 6.  Apparent  diurnal  motion  of  the  heavens. — As  the  earth 
revolves  from  west  to  east  on  the  axis  ns,  an   observer,  not 
being  conscious  of  this  motion,  sees  the  heavenly  bodies  appa- 
rently revolving  in  the  opposite  direction — that  is,  from  east  to 
west,  about  the  axis  NS.     The  sun,  moon,  and  every  planet, 


AND   DIURNAL   CIRCLES.  9 

comet,  and  star,  is  observed  to  pass  over  from  the  eastern  part 
of  the  sky  toward  the  western,  with  a  regular  motion,  reap- 
pearing again  in  the  east,  after  the  lapse  of  about  one  day,  in 
the  same,  or  nearly  the  same  place.  The  fixed  stars  describe 
circles,  which  are  exactly  parallel  to  the  equator,  and  in  pre- 
cisely the  same  length  of  time.  But  the  other  bodies  vary 
somewhat  in  their  paths,  and  the  periods  of  describing  them, 
thus  indicating  that  they  are  affected  by  other  motions  besides 
the  diurnal  rotation. 

17.  Rising,   setting,  and  culmination. — In   Fig.   3,  AB, 
DO,  FG,  etc.,  drawn  parallel  to  EQ,  represent  the  diurnal 
circles  of  stars,  projected  in   straight   lines.     Some   of  these 
circles  intersect  the  horizon  HO.     These  intersections  are  the 
points  of  rising  or  setting.     Thus,  a  star  describing  the  circle 
GF,  rises  in  the  northeast  quarter,  and  sets  in  the  northwest, 
at  points  which  are  both  represented  by  r.     The  star,  wThose 
diurnal  circle  is  IK,  rises  in  the  southeast,  and  sets  in  the  south- 
west, at  t.    A  star  on  the  equator  rises  exactly  in  the  east,  and 
sets  in  the  west,  at  the  point  C. 

The  points,  in  which  these  circles  cut  the  meridian,  are 
called  the  points  of  culmination.  Thus,  the  star  on  FG  makes 
its  upper  culmination  at  F,  and  its  lower  one  at  G.  On  AB, 
both  the  upper  and  lower  culminations  are  above  the  horizon  ; 
on  MP,  they  are  both  ~below.  If  both  culminations  of  a  star  are 
above  the  horizon,  it  is  always  in  view;  if  both  below,  it  never 
comes  in  sight.  The  number  of  stars  which  do  not  rise  and  set, 
depends  on  the  position  of  the  celestial  poles  in  relation  to  the 
horizon — that  is,  on  the  latitude  of  the  place. 

By  the  culmination  of  a  body,  in  the  ordinary  use  of  the 
word,  is  meant  its  upper  culmination. 

18.  Relations  of  the  horizon  to  the  diurnal    circles. — 
Every  change  of  position  on  the  earth  changes  the  horizon.     If 
an  observer  moves  eastward,  all  the  heavenly  bodies  which  rise 
and  set,  rise  earlier,  and  also  culminate  and  set  earlier.     If  he 
moves  westward,  they  rise,  culminate,  and  set  later.     If  he 
moves  toward  the  nearer  pole  of  the  earth,  the  corresponding 
pole  of  the  celestial  sphere  becomes  more  elevated,  and  the 


10  THE   PARALLEL  SPHERE. 

other  more  depressed  ;  and  the  contrary,  if  he  moves  from  the 
nearer  pole — that  is,  toward  the  equator.  In  all  north  latitudes, 
the  north  pole  is  elevated,  and  the  south  pole  depressed ;  and 
the  reverse  in  south  latitudes.  And  the  elevation  of  one  pole, 
and  the  depression  of  the  other,  equals  the  latitude.  For 
(Fig.  3)  E~O,  the  elevation  of  one  pole  (=HS,  the  depression  of 
the  other),  equals  EZ,  since  each  is  the  complement  of  ZN. 
But  EZ=£0,  the  latitude,  because  they  subtend  the  same  angle 
at  C. 

The  elevation  of  the  celestial  equator  equals  the  complement 
of  latitude.  For  EH  is  the  complement  of  EZ,  which  equals 
eo,  the  latitude.  Hence,  the  angle  by  which  all  the  circles  of 
diurnal  motion  are  inclined  to  the  plane  of  the  horizon,  equals 
the  complement  of  latitude,  since  they  are  parallel  to  the 
equator. 

On  account  of  this  change  of  inclination  between  the  horizon 
and  the  diurnal  circles,  the  aspect  of  the  diurnal  rotation  is 
very  different  in  different  parts  of  the  earth. 

19.  The  Tight  sphere. — This  name  is  given  to  those  posi- 
tions, in  which  the  diurnal  circles  cut  the  horizon  at  right 
angles.     All  points  of  the  equator  are  so  situated.     As  the 
latitude  is  zero,  the  poles,  having  no  elevation  or  depression 
(Art.  18),  are  both  in  the  horizon  ;  the  celestial  equator  passes 
through  the  zenith,  thus  coinciding  with  the  prime  vertical ; 
and  all  the  paths  of  daily  motion,  being  parallel  to  the  equator, 
are  perpendicular  to  the  horizon.    Every  heavenly  body,  unless 
situated  exactly  at  one  of  the  poles,  rises  and  sets  during  each 
revolution,  and  continues  above  the  horizon  just  as  long  as  it 
remains  below  it.     If  a  star  rises  in  the  east,  it  sets  in  the  west, 
and  culminates  in  the  zenith  and  nadir. 

20.  The  parallel  sphere. — This  term  expresses  the  appear- 
ance of  the  heavens  at  those  points  of  the  earth  where  the 
circles  of  daily  rotation  are  parallel  to  the  horizon.     This  aspect 
can  be  presented  only  at  the  poles.     For,  at  those  points,  the 
latitude  being  90°.  one  pole  must  be  elevated  90° — that  is,  to  the 
zenith — and  the  other  depressed  90°,  or  to  the  nadir.     Hence, 
the  diurnal  circles,  being  perpendicular  to  the  axis,  must  be 


ARTIFICIAL   GLOBES.  11 

horizontal,  and  the  equator  must  coincide  with  the  horizon. 
Every  star  in  view  parses  around  the  sky,  maintaining  the 
same  elevation  at  every  point  of  its  path.  No  one  of  the  fixed 
stars  ever  rises  or  sets,  and  every  point  of  a  diurnal  circle  may 
be  regarded  as  a  point  of  culmination,  since  it  is  on  a  meridian 
passing  through  the  observer's  place. 

At  the  north  pole,  that  half  the  year  in  which  the  sun  is 
north  of  the  equator,  is  uninterrupted  day ;  during  the  other 
half,  the  sun  being  south  of  the  equator,  it  is  constant  night. 

In  the  right  sphere,  the  whole  sky  is  seen,  and  every  part  of 
it  just  half  the  time  ;  in  the  parallel  sphere,  only  one-half  the 
sky  is  ever  seen,  but  it  is  seen  the  whole  time. 

2 1 .  The  oblique  sphere. — At  all  latitudes,  except  0°  and 
90°,  the  circles  of  daily  motion  are  oblique  to  the  horizon,  since 
they  incline  at  an  angle  equal  to  the  complement  of  the  lati- 
tude.    Thus,  at  42°  north  latitude,  the  celestial  equator  and  all 
the  diurnal  circles  are  elevated  48°  above  the  southern  horizon, 
as  represented  in  Fig.  3.     The  circle  OD,  whose  distance  from 
the  elevated  pole  equals  its  elevation,  just  touches  the  horizon 
at  the  lower  culmination,  and  is  the  limit  of  that  part  of  the 
sky  which  is  always  in  view.     This  is  called  the  circle  of  per- 
petual apparition.     The  circle  HL,  at  the  same  distance  from 
the  depressed  pole,  also  touches  the  horizon,  and  is  called  the 
circle  of  perpetual  occupation,  since  it  limits  that  part  of  the 
sky  which  is  always  concealed. 

The  horizon  HO,  bisects  the  equator  EQ.     Hence,  a  body 

on  the  equator  is  as  long  above  the  horizon  as  below  it,  in  every 

part  of  the  earth.     But  all  bodies  between  the  equator  and  the 

elevated  pole  are  longer  above  the  horizon  than  below,  while 

t  on  the  opposite  side  they  are  longer  below  than  above. 

22.  Artificial  globes. — They  are  of  two  kinds,  terrestrial 
and  celestial.     The  terrestrial  globe  is  a  miniature  representa- 
tion of  the  earth,  having  also  the  equator  and  several  meridians 
and  parallels  of  latitude  traced  upon  it.     The  celestial  globe 
exhibits  the  principal  fixed  stars  in   their  relations  to  each 
other,  and  to  the  equator  and  ecliptic. 

The  artificial  globe  is  suspended  in  a  strong  brass  ring  by  an 


12  PROBLEMS  ON  THE  GLOBES. 

axis  passing  through  the  north  and  south  poles,  on  which  it  is 
free  to  revolve.  This  ring  represents  the  meridian  of  any  place, 
and  is  supported  vertically  within  a  horizontal  wooden  ring 
which  stands  upon  a  tripod.  The  wooden  ring  represents  the 
horizon.  The  brass  ring  may  be  slid  around  in  its  own  plane, 
so  as  to  elevate  or  depress  either  pole  to  any  angle  with  the 
horizon.  It  is  graduated  from  the  equator  each  way  to  the 
poles,  for  measuring  latitude  and  declination  ;  while  the  horizon 
ring  has  near  its  inner  edge  two  graduated  circles,  one  for 
azimuth,  and  the  other  for  amplitude.  On  this  ring  also,  for 
convenient  reference,  are  delineated  the  signs  of  the  ecliptic, 
and  the  sun's  place  in  it  for  every  day  of  the  year. 

Around  the  north  pole  is  a  small  circle,  marked  with  the 
hours  of  the  day ;  and  at  the  same  pole,  a  brass  index  is  attached 
to  the  meridian,  which  can  be  set  at  any  hour  of  the  circle. 

The  quadrant  of  altitude  is  a  flexible  strip  of  brass,  graduated 
into  90  parts,  each  equal  to  a  degree  of  the  globe.  This  can 
be  used  for  measuring  angular  distances  in  any  direction  on  the 
sphere  ;  and  when  applied  to  a  vertical  circle  of  the  celestial 
globe,  it  determines  the  altitude,  or  zenith  distance  of  a  heav- 
enly body. 

To  adjust  either  globe  for  any  place  on  the  eartli,  elevate  the 
corresponding  pole  to  a  height  equal  to  the  latitude.  The  axis 
will  then  be  parallel  to  that  of  the  earth  or  the  heavens.  And 
if  the  globe  is  turned  (the  celestial  westward,  or  the  terrestrial 
eastward),  the  diurnal  motion  will  be  truly  represented. 

23.  Problems  on  the  terrestrial  globe. 

1.  To  find  the  latitude  and  longitude  of  a  place. 

Turn  the  globe  so  as  to  bring  the  place  to  the  brass 
meridian  ;  then  the  degree  and  minute  on  the  meridian 
over  the  place  shows  its  latitude,  and  the  point  of  the 
equator,  under  the  meridian,  shows  its  longitude. 

Example.  What  are  the  latitude  and  longitude  of 
New  York? 

2.  To  find  a  place  by  its  given  latitude  and  longitude. 

Find  the  given  longitude  on  the  equator,  and  bring 
it  to  the  meridian ;  then  under  the  meridian,  at  the 
given  latitude,  will  be  found  the  required  place. 


PROBLEMS  OK  THE  GLOBES.  13 

Ex.  What  place  is  in  latitude  39°  N.,  and  longitude 
77°  W.  ? 

3.  To  find   the  bearing  and  distance  of  one  place  from 
another. 

Adjust  the  globe  for  one  of  the  places,  and  bring  it 
to  the  meridian ;  screw  the  quadrant  of  altitude  directly 
over  the  place,  and  bring  its  edge  to  the  other  place^ 
Then  the  azimuth  will  be  the  bearing  of  the  second 
place  from  the  first,  and  the  number  of  degrees  between 
them,  multiplied  by  69J,  will  give  their  distance  apart 
in  miles. 

Ex.  Find  the  bearing  of  New  Orleans  from  New 
York,  and  the  distance  between  them. 

4.  To  find  the  difference  of  time  at  different  places. 

Bring  to  the  meridian  the  place  which  lies  west  of  the 
other,  and  set  the  hour-index  at  XII.  Turn  the  globe 
westward,  until  the  other  place  comes  to  the  meridian, 
and  the  index  will  show  the  hour  at  the  second  place 
when  it  is  noon  at  the  first.  The  hour  thus  found  is 
the  difference  required. 

Ex.  When  it  is  noon  at  New  York,  what  time  is  it 
at  London  ? 

5.  The  hour  being  given  at  any  place,  to  find  what  hour 
it  is  at  any  other  place. 

Find  the  difference  of  time  between  the  two  places, 
as  in  (4) ;  then,  if  the  place,  whose  time  is  required,  is 
east  of  the  other,  add  this  difference  to  the  given  time ; 
but  if  west,  subtract  it. 

Ex.  What  time  is  it  in  Boston,  when  it  is  2  p.  M.  in 
Paris  ? 

6.  To  find  the  antiscii,  the  perioeci,  and  the  antipodes  of  a 
given  place. 

Bring  the  given  place  to  the  meridian ;  then,  under 
the  meridian,  in  the  opposite  hemisphere,  in  the  same 
degree  of  latitude,  are  found  the  antiscii.  Set  the 
index  to  XII.,  and  turn  the  globe  until  the  other  XII. 
is  Tinder  the  index ;  then,  the  periceci  will  be  at  the 
same  point  of  the  meridian  as  the  given  place  was,  and 
the  antipodes  will  be  where  the  antiscii  were. 


PROBLEMS   OX   THE   GLOBES. 

Ex.  Find  the  antiscii,  the  perioeci,  and  the  antipodes 
of  Lake  Superior. 

To  the  antiscii,  the  hour  of  the  day  is  the  same  as  at 
the  given  place,  but  the  season  is  reversed.  To  the 
perioeci,  the  season  is  the  same,  but  the  hour  opposite. 
To  the  antipodes,  both  hour  and  season  are  opposite. 

24.  Problems  on  the  celestial  glole. 

1.  To  find  the  right  ascension  and  declination  of  a  heav- 
enly body. 

Bring  the  place  of  the  body  to  the  meridian ;  then 
the  point  directly  over  it  shows  its  declination ;  and  the 
point  of  the  equator  under  the  meridian,  its  right 
ascension. 

Ex.  Find  the  right  ascension  and  declination  of  a 
Lyrse.  Also,  of  the  sun  on  the  3d  of  May. 

2.  To  represent  the  appearance  of  the  heavens  at  any  time. 

Adjust  the  globe  for  the  place.  (Art.  22.)  On  the 
wooden  horizon  find  the  day  of  the  month,  and  against 
it  is  given  the  sun's  place  in  the  ecliptic.  On  the 
ecliptic  find  the  same  sign  and  degree,  and  bring  the 
point  to  the  meridian.  The  globe  then  presents  the 
positions  of  the  stars  at  noon.  Set  the  hour-index  at 
XII.,  and  turn  the  globe  till  the  index  points  to  the 
required  hour.  The  aspect  of  the  heavens  at  that  hour 
is  then  represented. 

Ex.  Required  the  aspect  of  the  stars  at  Lat.  51°,  Dec. 
5th,  at  10  P.  M. 

3.  To  find  the  time  of  the  rising  and  setting  of  any  heav- 
enly body,  at  a  given  place. 

Having  adjusted  for  the  latitude,  bring  the  sun's 
place  in  the  ecliptic  to  the  meridian,  and  set  the  index 
at  XII.  Turn  the  globe  eastward,  and  then  westward, 
till  the  given  body  meets  the  horizon,  and  the  index 
will  show  the  times  of  rising  and  setting. 

The  times  of  the  suns  rising  and  setting  may  be 
found  in  the  same  manner,  on  the  terrestrial  globe, 
since  the  ecliptic  is  usually  represented  on  it. 


PARALLAX  DEFINED.  15 

Ex.  At  what  time  does  the  sun  rise  and  set  on  the 
4th  of  July  ? 

Find  the  time  of  the  rising  and  setting  of  Arcturus 
on  the  10th  of  November. 

4.  To  find  the  altitude  and  azimuth  of  a  star  for  a  given 
latitude  and  time. 

Adjust  the  globe  for  the  aspect  of  the  heavens  (2); 
screw  the  quadrant  of  altitude  to  the  zenith,  and  direct 
it  through  the  place  of  the  star.  Then,  the  arc  between 
the  star  and  the  horizon  is  the  altitude ;  and  the  arc  of 
the  horizon  between  the  quadrant  of  altitude  and  the 
meridian,  is  the  azimuth. 

Ex.  Find  the  altitude  and  azimuth  of  Sirius,  Dec. 
25th,  at  9  P.  M.  Lat.  43°. 

5.  To  find  the  angular  distance  between  two  stars. 

Lay  the  quadrant  of  altitude  across  the  two  stars,  so 
that  the  zero  shall  fall  on  one  of  them  ;  then,  the  degree 
at  the  other  will  show  their  distance  from  each  other. 

Ex.  Find  the  distance  between  Arcturus  and  a  Lyras. 

6.  To  find  the  sun's  meridian  altitude  for  a  given  latitude 
and  day. 

Find  the  sun's  place,  and  bring  it  to  the  meridian. 
The  degree  over  it  will  show  its  decimation.  If  the 
declination  and  latitude  are  both  north  or  south,  add 
the  declination  to  the  co-latitude  ;  if  not,  subtract  it. 

Ex.  Find  the  sun's  meridian  altitude  at  noon,  Aug. 
1st.  Lat.  38°  30'  K  - 


CHAPTEE  II. 

PARALLAX. — ATMOSPHERIC   REFRACTION. — TWILIGHT. 

25.  Parallax  defined. — When  a  person  changes  his  place, 
objects  about  him  in  general  appear  in  different  directions  from 
him.  This  change  of  direction  is  called  parallax.  If,  for  ex- 
ample, he  moves  north,  an  object,  which  was  directly  west  of 


16 


DIURNAL   PARALLAX. 


him,  is  moved  by  parallax  towards  the  southwest  /  and  an 
object  which  was  east,  now  appears  in  the  southeast  quarter. 
The  direction  of  every  thing  is  more  or  less  altered,  except 
those  objects  which  are  in  the  line  of  his  motion. 

26.  Diurnal  parallax.  —  While  a  person  therefore  travels 
over  the  earth,  or  is  carried  about  it  by  the  diurnal  rotation, 
the  heavenly  bodies  must  in  the  same  way  suffer  some  paral- 
lactic  change. 

By  the  true  place  of  a  heavenly  body,  is  meant  that  which  it 
would  seem  to  occupy  if  viewed  from  the  center  of  the  earth. 
At  the  surface,  therefore,  it  appears  generally  displaced  from 
its  true  position  ;  and  this  displacement  is  called  the  diurnal 
parallax.  Thus,  the  true  place  of  the  body  M  (Fig.  4.),  is  in 
the  direction  CK  ;  but  at  A  it  appears  in  the  line  AH  ;  and  the 
parallax  is  the  angle  AMC. 
So,  the  true  place  of  M'  is  Q, 
its  apparent  place  is  P,  and 
the  parallax  is  AM'C.  But 
the  body  Wf  appears  at  Z, 
whether  viewed  from  A  or  C, 
and  the  parallax  in  this  case  is 
zero.  Since  the  earth's  radius, 
in  each  instance,  subtends  the 
angle  of  parallax,  we  have  the 
following  definition  : 

The  diurnal  parallax  of  a 
body  is  the  angle  at  that  body 
subtended  l>y  the  semi-diameter  of  the  earth. 

27.  On  what  diurnal  parallax  depends.  —  In  the  triangle 
ACM',  let  AC=r,  CM'=<Z,  and  the  parallax.  AM'C=p.     Let 
the  zenith  distance  of  the  body,  ZAM'  =  &  ;  then,  the  angle 
CAM'  is  the  supplement  of  s.     Hence, 

sin  &   :  :   r   :   d\ 


Fig.  4. 


R 


Q 


K 


r  sn  z 

...  sinjp  =  __ 

Since  p  is  always  very  small,  sin  p  varies  nearly  as^>  itseif 


PARALLAX  OF   THE    MOON.  17 

Therefore,  regarding  r  as  constant,  p  cc  —-*—•     That  is,  The 

parallax  of  a  lody  varies  directly  as  the  sine  of  its  zenith 
distance,  and  inversely  as  its  distance  from  the  earth. 

28.  Horizontal  parallax. — The   largest    diurnal  parallax, 
which  a  body  can  have,  occurs  when  the  body  is  seen  in  the 
horizon,  as  at  M.     It  is  then  called  horizontal  parallax.   From 
the  horizon  to  the  zenith,  the  parallax  diminishes  through  all 
values  to  zero. 

In  the  case  of  a  given  body,  d  is  usually  constant ;  and  if  its 
parallax,  at  a  certain  elevation,  has  been  obtained,  its  horizontal 
parallax  is  found  by  the  variation,  p  oo  sin  z.  At  the  horizon, 
z  =  90°,  and  sin  z  —  rad.  If,  when  the  zenith  distance  is  53°, 
the  moon's  parallax  is  found  by  observation  to  be  45',  then 
sin  53°  :  rad  : :  45'  :  56'  2V,  which  is  its  horizontal  parallax. 

29.  To  correct  for  parallax. — The  effect  of  parallax  is  to 
cause  a  body  to  appear  lower  than  its  true  place.     Hence,  the 
true  altitude  of  a  body  is  obtained  by  adding  the  parallax  to 
its  apparent  altitude. 

As  parallax  is  a  depression  on  a  vertical  circle,  then,  .if  a 
body  is  on  the  meridian,  the  parallax  affects  its  declination  just 
as  much  as  its  altitude,  since  the  meridian  is  also  a  vertical ;  but 
in  other  cases,  the  vertical  circle  being  oblique  to  the  equator, 
the  parallax  can  be  resolved  into  two  components,  one  of  which, 
parallel  to  the  equator,  is  parallax  in  right  ascension  ;  the  other, 
peripendicular  to  the  equator,  is  parallax  in  declination. 

30.  To  find  the  parallax  of  the  moon. — Let  A  and   B 
(Fig.  5)  be  two  stations  on  the  same  meridian,  taken  as  far 
apart  as  possible.     The  latitude  of  each  place  being  known,  the 
arc  AB — that  is,  the  angle  ACB — is  known.     When  the  moon 
crosses  the  meridian,  let  its  zenith  distance  be  observed  at  each 
station.     The  observer  A  sees  the  moon  projected  in  the  sky  at 
Y,  and  the  zenith  distance  is  the  angle  ZAY,  while  that  at  B 
is  Z'BY'.     The  supplements  of  these  angles,  MAC,  MBC,  are 
therefore  known.     In  the  isosceles  triangle  ABC,  obtain  the 
angles  A  and  B,  and  the  side  AB ;  subtract  the  angles  from 


18 


ATMOSPHERIC    REFRACTION. 


Fig.  5. 


Z 


MAC  and  MBC  respectively,  then  MBA,  MAB  are  known, 
which,  with  the  side  AB,  will  give  AM  and  BM.  Finally,  in 
the  triangle  AMC,  the  angle  A  and  sides  including  it  will  fur- 
nish the  angle  AMC,  which  is  the  parallax  sought  for  the 
station  A,  at  the  zenith 
distance  ZAY.  From 
this  the  .Horizontal  paral- 
lax can  be  obtained,  as  in 
Art.  28. 

The  horizontal  paral- 
lax of  the  moon  is  much 
greater  than  that  of  any 
other  heavenly  body.  Its 
mean  value  is  about  57', 
and  is  correctly  repre- 
sented by  the  angle 
EMC,  in  Fig.  6. 

The  above  method  has 
also  been  employed  for 
two  or  three  of  the 
-planets,  when  they  come  near  to  the  earth.  But,  with  these 
exceptions,  all  the  heavenly  bodies  are  so  far  from  us,  that  their 
horizontal  parallax  is  too  small  to  be  obtained  in  this  way  with 
sufficient  accuracy.  The  parallax  of  the  sun  is  less  than  9" ; 
that  of  nearly  all  the  planets  is  much  smaller  than  this ;  and  as 
to  bodies  outside  of  the  solar  system,  they  afford  not  the 
slightest  indication  of  any  diurnal  parallax. 

Fig.  6. 


31.  Atmospheric  refraction. — Before  the  true  place  of  a 
body  can  be  found  by  observation,  a  correction  must  also  be 
applied  for  the  refraction  of  its  light  by  the  atmosphere.  While 
parallax  depresses  bodies  below  their  true  places,  more  or  less 
according  to  their  distance,  refraction  elevates  them,  the  near 
and  the  distant  alike. 

The  earth's  atmosphere  may  be  conceived  to  consist  of  an 


ATMOSPHERIC   REFRACTION. 


19 


indefinite  number  of  strata,  bounded  by  spherical  surfaces,  as 
AA,  BB,  etc.  (Fig.  7),  these  strata  being  more  dense  according 
as  they  are  nearer  the  earth.  Light  from  a  star  S,  entering  the 
air  at  a,  is  bent  toward  the  perpendicular  to  its  surface  (which 

Fig.  7. 


is  the  earfh's  radius  produced  to  that  point),  and  describes  ab, 
instead  of  ax.  For  the  same  reason,  it  is  again  bent  into  be, 
and  then  into  cO  ;  and  therefore  the  star  appears  in  the  direc- 
tion of  cO  produced,  at  S',  higher  than  its  true  place.  The 
path  of  the  ray  from  a  to  O  is  in  reality  not  a  broken  line,  as 
in  the  figure,  but  a  curve,  because  the  changes  of  density  occur 
at  every  point.  A  body  at  the  zenith  is  not  moved  out  of 
place,  because  its  light  strikes  the  surfaces  perpendicularly. 
The  refraction  at  the  horizon  is  about  3-4'.  This  is  the  greatest 
of  all,  since  the  angle  of  incidence  there  is  the  greatest  possible. 
From  the  zenith  to  the  horizon  the  refraction  constantly  in- 
creases,— slowly  at  great  elevations,  but  very  rapidly  near  the 
horizon,  as  shown  in  the  following  table. 


Elevation. 

Eefraction. 

Elevation. 

Eefraction. 

90° 

0'    0" 

20° 

2'  39" 

80 

0   10 

10 

5   20 

60 

0   33 

5 

10   00 

45 

0   58 

2 

18   00 

40 

1    09 

1 

24   25 

30          i       1   40         S 

0 

34   30 

The  true  size  of  the  largest  angle  of  refraction  is  seen  in 


20  METHODS  OF   MEASURING   REFRACTION. 

Fig.  8.  AB  is  a  portion  of  the  surface  of  the  earth,  db  the 
surface  of  the  atmosphere,  AC,  BC  portions  of  the  radii  of  the 
earth ;  S  is  the  true  place  of  a  star,  S'  the  place  as  elevated  hy 
horizontal  refraction. 


a 


A 


Fig.  8. 

F 
"s 


32.  Measurement   of    refraction. — At    latitudes   greater 
than  45°,  stars  which   culminate  in   the  zenith  make  their 
lower  culminations  above  the  horizon.     Such  a  star  is  observed 
at  both  culminations,  and  its  distance  from  the  pole  is  measured 
at  each.     These  polar  distances  are  really  equal,  but  appa- 
rently unequal,  because  below  the  pole  the  star  is  elevated  by 
refraction,  while  at  the  zenith  it  is  not  displaced.     The  differ- 
ence  of  the   apparent  polar  distances,   therefore,    gives  the 
amount  of  refraction  at  the  place  of  lower  culmination. 

The  refraction  within  several  degrees  of  the  zenith  is  so 
slight,  and  its  change  so  uniform,  that  observations  may  be 
made  in  the  same  way  on  stars  which  culminate  several  degrees 
north  or  south  of  the  zenith ;  and  thus,  by  applying  a  small 
correction,  the  refraction  may  be  measured  at  many  different 
altitudes. 

33.  General   method   of  measuring    refraction. — A    star, 
whose  declination  is  known,  may  be  used  for  determining  re- 
fraction at  any  altitude,  in  the  following  manner. 

Let  m  n  (Fig.  9)  be  the  path  of  diurnal  rotation  of  a  star, 
whose  declination  xr  is  known.  When  the  star  is  at  a?,  let  its 
apparent  altitude,  xy,  be  measured,  and  the  exact  time  also  be 
observed.  When  it  culminates  at  m,  observe  the  time  again. 
The  difference  of  these  times,  allowing  15°  for  an  hour,  will 
give  the  angle  at  the  pole  ZPx.  The  co-latitude  of  the  place, 
ZP,  and  the  co-declination  of  tjie  star,  Pa?,  being  known  in  the 


TABLES  OF   REFEACTION. 


21 


spherical  triangle  ZP#,  the  side  Za?  can  be  computed.  Its 
complement  xy  is  the  true  altitude.  This,  subtracted  from  the 
apparent  altitude  before  observed,  gives  the  refraction  at  that 
elevation. 

Fig.  9. 


E 


34.  Tables  of  refraction. — It  is  demonstrated,  that  except 
near  the  horizon,  the  mean  refraction  varies  as  the  tangent  of 
the  zenith  distance.     Tables  of  atmospheric  refraction  are  cal- 
culated in  accordance  with  this  law,  for  all  zenith  distances 
less  than  80°.     They  are,  however,  extended  beyond  that  limit 
down  to  the  horizon,  being  calculated  for  the  last  10°  by  a 
different  and  more  complex  law,  and  the  results  of  calculation 
being   more    uncertain.      On   this  account,   all    astronomical 
measurements  are  made,  so  far  as  is  possible,  within  75°  of  the 
zenith.     In  order  to  obtain  the  place  of  a  body  with  the  utmost 
accuracy,  tables  of  refraction  are  .accompanied  with  means  of 
correcting  for  the  state  of  the  barometer  and  the  thermometer 
at  the  time  of  observation. 

35.  Time   of  rising  and  setting  affected  by  refraction. — 
Since  any  heavenly  body  at  the  horizon  is  considerably  elevated 
by  refraction,  it  therefore  appears  to  rise  earlier  and  set  later 


22  TWILIGHT. 

than  it  would  do  if  there  were  no  atmosphere.  The  angular 
breadth  of  the  sun  is  about  32',  while  horizontal  refraction  is  a 
little  more  than  this— 34  J'.  Therefore,  the  sun  appears  just 
above  the  horizon,  when,  in  truth,  it  is  wholly  below.  This 
adds  at  least  four  minutes  to  the  day,  two  in  the  morning  and 
two  at  evening. 

36.  Distortion  of  the  surfs  and  moon's  disk   by  refrac- 
tion.— The  change  in  the  amount  of  refraction  is  so  rapid  near 
the  horizon,  that  when  the  sun  has  just  risen,  or  is  just  about  to 
set,  the  lower  limb  is  elevated  more  than  the  upper,  by  a  very 
perceptible  quantity.     Its  form,  therefore,  does  not  appear  cir- 
cular, but  nearly  elliptical,  the  vertical  diameter  being  shortened 
about  5'  or  6'. "  The  lower  half,  however,  appears  more  flat- 
tened than  the  upper  half,  because  the  difference  of  refraction 
between  the  lower  limb  and  the  center  is  greater  than  that 
between  the  center  and  the  upper  limb. 

37.  Illumination  of  the  sky. — During  the  day,  the  atmos- 
phere is  illuminated  by  the  light  of  the  sun,  which  penetrates 
every  part  of  it,  and  is  reflected  in  all  directions.     If  there  were 
no  air,  the  sky,  instead  of  appearing  luminous  by  day,  would 
exhibit  the  same  blackness  as  by  night,  and  the  stars  would  be 
visible  alike  at  all  times.     We  should,  in  that  case,  lose  a  great 
part   of  that  generally  diffused   light  which  illuminates  the 
interior  of  buildings,  and  other  places  screened  from  the  direct 
rays  of  the  sun.     The  earth's  surface,  and  all  terrestrial  objects, 
on  which  the  sunlight  falls  directly,  would  indeed,  by  radiant 
reflection,  cause  a  degree  of  illumination,  but  it  would  be  far 
less  than  we  now  enjoy.     It  has  been  observed,  in  ascending  to 
great  heights,  either  on  mountains  or  in  balloons,  where,  of 
course,  the  air  which  is  most  dense  and  reflects  most  abun- 
dantly is  left  below,  that  the  sky  assumes  a  very  dark  hue,  and 
the  general  illumination  is  greatly  diminished. 

38.  Twilight. — The  illumination  of  the  sky  begins  before 
the  sun  rises,  and  continues  after  it  sets :  it  is  then  called  twi- 
light.    More  or  less  of  it  is  visible,  as  long  as  the  sun  is  not 
more  than  1 8°  vertically  below  the  horizon.     Those  parts  of  the 


DURATION   OF  TWILIGHT.  23 

atmosphere  are  most  luminous,  which  lie  nearest  to  the  direc- 
tion of  the  sun.  Thus,  in  Fig.  10,  let  A  be  a  place  on  the 
earth,  where  the  sun  is  just  setting.  The  whole  sky,  IEFH,  is 
illuminated.  But,  to  a  place  further  east,  as  B,  the  twilight 
extends  from  E  to  H, — the  part  of  the  sky,  HK,  remote  from 
the  sun,  being  in  the  shadow  of  the  earth.  At  C,  only  FH  is 
illuminated,  and  HL  is  dark.  At  D,  the  twilight  is  entirely 
gone. 

Fig.  10. 
H 


Though  the  twilight  terminates  at  H,  there  is  no  abrupt 
transition  from  light  to  shade  at  that  point,  since  the  reflection 
from  those  high  and  rare  parts  of  the  air  is  exceedingly  feeble  ; 
and  also,  because  the  thickness  of  the  illuminated  segment, 
through  which  we  look,  diminishes  gradually  to  that  limit,  as 
is  obvious  from  an  inspection  of  the  figure. 

39.  Duration  of  twilight. —  To  an  observer  at  the  equator, 
at  those  times  of  the  year  when  the  sun  is  on  the  celestial 
equator,  the  twilight  continues  Ih.  12ni.  For,  in  the  diurnal 
motion,  15°  are  described  in  an  hour,  and  therefore  18°  in 
ly^h.  =  Ih.  12m.  This  is  the  shortest  duration  possible.  For, 
if  the  sun  were  on  a  parallel  of  declination,  the  degrees  of  diurnal 
motion  would  be  shorter  than  those  on  a  great  circle.  And, 
if  the  observer  were  on  some  parallel  of  latitude,  the  circles  of 
daily  motion  would  be  oblique  to  his  horizon,  and  the  sun  must 
therefore  pass  over  more  than  18°,  in  order  to  move  18°  verti- 
cally. An  extreme  case  occurs  at  the  poles,  where  twilight 
lasts  several  months. 


THE   TRANSIT   INSTRUMENT. 


CHAPTER    III. 

THE   OBSERVATORY   AND    ITS   INSTRUMENTS.— SPHERICAL 
PROBLEMS. 

40.  The  observatory. — Accurate  knowledge  of  the  motions 
of  the  heavenly  bodies  is  mostly  obtained  by  observing  their 
relations  to  the  diurnal  rotation.     The  observatory  is  furnished 
with  several  instruments  by  which  such  observations  are  made. 

41.  The  transit  instrument. — This  is  a  telescope  so  mount- 
ed as  to  observe  a  heavenly  body,  at  the  instant  when  it  cul- 
minates— that  is,   makes   a  transit    of  the    meridian.     AD 


Fig.  11. 


(Fig.  11)  represents  the  telescope  supported  by  a  horizontal 
axis,  which  consists  of  two  hollow  cones,  placed  base  to  base, 
so  as  to  combine  lightness  and  strength.  The  ends  of  the  axis 
rest  in  sockets,  attached  to  two  stone  piers,  E  and  "W.  That 


ADJUSTMENT   OF   TRANSIT   INSTRUMENT. 


25 


the  instrument  may  receive  no  tremors  from  the  building,  the 
piers  stand  on  a  firm  foundation  in  the  ground,  passing  through 
the  floor  without  contact.  The  axis  being  placed  east  and 
west  horizontally,  the  telescope,  which  is  perpendicular  to  it, 
will,  when  turned,  revolve  in  the  plane  of  the  meridian.  A 
graduated  circle,  n,  is  attached  to  one  end  of  the  axis,  for 
marking  altitudes  or  zenith  distances.  The  whole  instrument 
can  be  raised  from  the  sockets,  and  the  axis  inverted,  so  that 
the  east  end  shall  rest  on  the  pier  W,  and  the  west  end  on  the 
pier  E. 

42.  Adjustments  of  the  transit  instrument. — The  visual 
axis  of  the  telescope,  AD,  is  called  the  line  of  collimation^  and 
is  marked  by  the  intersection  of  two  exceedingly  fine  wires  in 
the  focus  of  the  eye-glass.  One  of  these  wires  is  horizontals/A 
(Fig.  12),  the  other  vertical,  de\  the  latter  visibly  marks  the 
direction  of  the  meridian,  when  the  instrument  has  been  prop- 
erly adjusted.  The  sockets,  in  which  the  ends  of  the  axis 
rest,  are  so  connected  with  the  stone  piers,  that  one  of  them  can 
be  raised  or  lowered  by  a 
screw,  and  the  other  can,  in 
a  similar  manner,  be  moved 
north  or  south.  By  the 
spirit-level,  L,  which  hangs 
on  the  axis,  it  can  be  seen 
whether  the  axis  is  horizon- 


Fig.  12. 


tal.  If  not,  raise  or  lower 
the  end  which  admits  of 
vertical  motion.  To  find 
whether  the  line  of  collima- 
tion  is  perpendicular  to  the 
axis  of  revolution,  observe 

9 

whether  a  distant  terrestrial 

cbject,  which  is  on  the  vertical  wire,  remains  on  it  after  the 
ends  of  the  axis  have  been  inverted  in  their  sockets.  If  not, 
move  the  plate  which  carries  the  wires  laterally,  till  the  vertical 
wire  bisects  the  distance  between  the  two  positions  of  the 
object.  And  finally,  to  determine  whether  the  axis  is  east  and 
west,  observe  if  a  circumpolar  star  occupies  the  same  length  of 


26  TO   OBSERVE   EIGHT  ASCENSION. 

time  in  passing  from  the  upper  to  the  lower  culmination,  as 
from  the  lower  to  the  upper ;  and  if  not,  move  the  end  of  the 
axis  horizontally,  till  the  intervals  are  equal. 

For  fuller  instructions  on  adjustment,  see  Loomis's  Practical 
Astronomy. 

43.  The  astronomical  clock. — The  transit  instrument  marks 
the  event  of  crossing  the  meridian ;  the  clock  must  be  used  in 
connection  with  it,  to  fix  the  time  of  the  transit.     The  clock  of 
the  observatory  is  made  to  keep  sidereal  time, — that  is,  it  marks 
off  24  hours  in  the  interval  between  two  successive  transits  of  a 
star,  instead  of  the  sun.     This  interval  is  called  a  sidereal  day, 
and  is  about  4  minutes  less  than  a  solar  day.     The  sidereal  day 
begins  when  the  vernal  equinox  transits  the  meridian.     At  that 
instant,  the  clock  is  at  Oh.  Om.  Os. ;  and  any  hour  of  the  clock 
shows  how  long  a  time  has  elapsed  since  the  equinox  culmi- 
nated. 

44.  Error  and  rate  of  clock. — The  uniform  movement  of 
the  clock  is  its  most  important  excellence.     This  may  be  tested 
by  the  transit  instrument,  and  a  list  of  right  ascensions  of  stars. 
If  it  does  not  indicate  Oh.  Om.  Os.  when  the  vernal  equinox  cul- 
minates, the  difference  is  called  its  error.     If  it  marks  any  more 
or  less  than  24  hours  between  two  successive  transits  of  a  star, 
this  gain  or  loss  is  called  its  rate.     If  both  error  and  rate  are 
known,  then  the  true  time  is  known ;  and  generally  it  is  not 
best  to  alter  the  clock,  but  only  to  keep  a  record  of  error  and 
rate. 

45.  To  observe  the  right  ascension  of  a  heavenly  lody. — 
Having  elevated  the  telescope  to  the  altitude  of  the  body  at  the 
time  of  culmination,  notice  the  exact  instant  when  it  appears 
on  the  vertical  wire  de  (Fig.  12).     This  is  its  right  ascension, 
which  may  be  given  either  in  time  or  in  arc.     Thus,  if  the 
clock  is  at  13h.  46m.  32s.  when  a  star  passes  the  wire,  its  right 
ascension  is  13h.  46m.  32s. ;  or,  at  the  rate  of  15°  for  each 
hour,  206°  38'  0". 

To  secure  greater  accuracy,  several  equidistant   wires   are 
placed  parallel  to  de,  an  equal  number  on  each  side,  as  in  Fig. 


THE   CHEONOGEAPH.  27 

12.  The  time  of  passing  each  wire  is  noted,  and  the  average  of 
all  obtained  for  the  time  of  crossing  the  central  one. 

To  observe  the  right  ascension  of  the  sun  or  a  planet,  the 
transit  of  each  limb  must  be  noticed,  and  the  mean  of  all  the 
times  will  be  the  right  ascension  of  the  center  of  the  disk. 

In  order  to  render  the  wires  visible  by  night,  the  field  of 
view  is  faintly  illuminated  by  a  lamp,  placed  at  one  end  of  the 
hollow  axis,  the  light  of  which,,  after  entering  the  telescope,  is 
reflected  toward  the  eye-piece. 

46.  Transits  recorded  1y  the  chronograph. — To  observe  the 
time  of  a  star-transit,  the  eye  must  discern  the  instant  of  its 
bisection  by  the  wire,  and  the  ear  must  hear  the  beat  of  the 
clock, — the  seconds  being  counted  from  the  last  completed 
minute  before  the  observation  began.  If  the  bisection  occurs 
between  two  beats,  as  it  commonly  does,  the  observer  needs 
much  practice  to  be  able  to  divide  the  second  accurately  into 
tenths,  and  decide  at  which  of  them  the  transit  takes  place. 
Transits  are  now  generally  observed  and  recorded  with  much 
greater  ease  and  accuracy  by  the  use  of  the  galvanic  circuit. 

Fig.  13. 


The  pendulum  of  the  observatory  clock  is  arranged  to  close 
the  circuit  of  a  battery  and  break  it  again,  at  the  beginning  of 
every  beat.  The  closing  of  the  circuit  gives  a  small  lateral 
motion  to  the  registering  pen,  under  which  the  paper  is  ad- 
vancing on  a  revolving  cylinder,  about  an  inch  per  second. 
Thus  the  seconds  are  all  permanently  recorded  by  notches  one 
inch  asunder  in  a  straight  line,  as  «,  &,  c,  d  (Fig.  13).  The 
mark  at  the  beginning  of  each  minute  has  some  peculiarity  by 
which  it  may  be  distinguished  from  the  rest.  The  observer  has 
under  his  hand  a  key,  which,  by  a  quick  touch,  will  also  close 
and  break  the  circuit.  Whenever  a  star  is  on  one  of  the  wires 
of  the  transit  instrument,  he  touches  the  key,  the  pen  is  moved 
aside,  and  indents  the  line  as  at  A,  and  the  observation  is  thus 
recorded ;  and  the  place  where  this  motion  commenced  between 
the  second-marks  can  afterward  be  carefully  examined.  Thus, 


28 


THE   MURAL   CIRCLE. 


without  the  distraction  of  attending  to  the  clock,  he  can  record 
the  transits  of  all  the  wires  ;  and  if  he  only  notices  within  what 
•minute  the  work  begins,  he  can  read  the  entire  record  with 
accuracy  to  the  j^  or  even  the  T^o  of  a  second.  Since  the 
general  adoption  of  this  method,  the  number  of  wires  has  been 
increased,  sometimes  to  30  or  40,  so  as  to  obtain  the  mean  of 
more  numerous  observations  on  the  same  star.  The  instru- 
ment, as  above  described,  is  known  as  the  chronograph. 

47.  The  mural  circle. — The  circle  of  the  transit  instrument 
is  used  principally  for  finding  a  body  whose  altitude  is  known, 
and  is  too  small  for  accurate  measurement  of  arcs  on  the  meri- 

Fig.  14. 


dian.  For  measuring  meridian  arcs,  the  mural  circle  is  em- 
ployed ;  so  called,  because  it  revolves  by  the  side  of  a  vertical 
wall.  It  consists  of  a  circle  usually  six  or  eight  feet  in  diame- 
ter, and  a  telescope  attached  to  its  face.  It  is  made  so  large,  in 


THE   VERNIEK. 


29 


order  that  very  small  angles  may  be  measured  by  the  divisions 
on  its  limb.  Fig.  14  represents  the  instrument  attached  to  the 
meridian  wall.  Its  radii  are  hollow  and  of  conical  form.  The 
axis,  which  is  on  one  side  only,  is  firmly  set  in  the  wall ;  and 
the  circle  and  telescope  revolve  upon  it.  The  graduations  are 
made  on  the  rim,  and  not  on  the  face  of  the  circle,^  and  are  read 
by  means  of  microscopes  attached  to  the  wall. 

48.  Subdivisions  of  the  graduated  limb. — The  reading  of  a 
graduated  arc  can  always  be  carried  much   lower  than   the 
divisions  actually  marked  on  it.     This   is  sometimes  accom- 
plished by  the  vernier,  and  sometimes  by  the  reading  micro- 
scope. 

49.  The  vernier. — This  contrivance,  so  named  from  the  in- 
ventor, is  a  short  graduated  arc,  which  slides  along  the  limb  of 
the  circle  that  is  to  be  subdivided.     For  example,  AB  (Fig.  1 5) 
is  a  vernier  for  dividing  the  12'  spaces  of  the  arc  on  its  right 
into  portions  of  V  each.     For  this  purpose,  the  vernier  consists 
of  12  parts,  which  together  are  equal  to  11  of  the  divisions  of 
the  limb.     Since  12  parts  of  the  vernier  are 

less  than  12  divisions  of  the  arc  by  a  whole 
division,  one  part  of  the  vernier  is  less  than  one 
division  of  the  arc  by  y1^  of  a  division ;  two  are 
less  than  two  by  T2^  °f  a  division,  and  so  on. 
Now,  in  the  figure,  the  zero  of  the  vernier  has 
passed  10°  24' ;  and  in  order  to  find  how  many  1( 
twelfths  of  the  next  space  it  has  passed,  it  is  s 

only  necessary  to  look  along  the  vernier,  and 
observe  the  number  of  the  division  line,  which 

/» 

coincides  with  a  line  of  the  arc.     In  this  case  , 

we  find  it  to  be  ls|ie  8th.     Hence,  the  8  parts  4 

of  the  vernier  from  0  to  8  are  less  than  the  cor-  « 

responding  8  divisions  of  the  arc  by  T8^ ;  that 
is,  zero  is  T8^  of  12'  beyond  10°  24'.     Therefore          Q 
the  reading  is  10°  32'. 

The  vernier  is  sometimes  made,  so  that  a 
given  number  of  parts  equals  one  more,  instead 
of  one  less,  than  the  same  number  on  the  limb. 
But  the  principle  of  making  subdivisions  is  the  same. 


Fig.  15. 


-13° 


-12° 


-11 


30  TO   FIND   DECLINATION. 

50.  The  reading  microscope. — This  is  a  compound  micro- 
scope, having  in  the  focus  of  its  eye-piece  a  pair  of  spider-lines 
intersecting  each  other,  and  in  the  same  field  of  view  are  the 
magnified  divisions  of  the  arc.     The  intersection  of  the  spider- 
lines  is  moved  laterally  from  one  division  line  of  the  arc  to 
another  by  a  screw.     If  the  divisions,  for  example,  are  equal  to 
5'  each,  then  the  screw  is  so  made  as  to  move  the  intersection 
from  one  line  to  another  by  five  revolutions,  and  therefore  each 
revolution  indicates  a  motion  of  V.     A  circle  is  attached  to  the 
axis  of  the  screw,  having  its  circumference  divided  into  60 
equal  parts.     As  each  revolution  of  the  screw  can  thus  be 
divided  into  60  equal  parts,  so  each  minute  of  the  arc  can  be 
divided  into  seconds. 

One  of  these  reading  microscopes  is  represented  at  A  (Fig. 
14) ;  and  the  places  of  others  are  marked  at  B,  C,  D,  E,  F,  60° 
from  each  other.  Six  are  used,  instead  of  one,  for  the  purpose 
of  obtaining  a  more  accurate  result,  by  taking  a  mean  of  the 
seconds  in  the  several  readings. 

51.  To  find  the  declination  of  a  heavenly  body. — This  may 
be  done  by  measuring  its  meridian   altitude.     Let  the  mural 
circle  be  adjusted  in  altitude,  so  that,  at  the  instant  when  the 
body  crosses  the  vertical  wire  of  the  telescope,  it  is  on  the 
horizontal  wire  also.     The  graduation  of  the  limb  shows  its 
altitude.     The  latitude  of  the  observatory  being  known,  the 
elevation  of  the  equator  is  known  ;  and  the  difference  between 
the  altitude  of  the  body  and  the  elevation  of  the  equator,  is  the 
declination  sought.     In  northern  latitudes,  if  the  altitude  of  the 
heavenly  body  exceeds  the  elevation  of  the  equator,  the  differ- 
ence is  a  northern  declination ;  if  it  is  less,  the  declination  is 
south.  $ 

Before  altitudes  can  be  measured,  the  horizontal  position  of 
the  telescope  must  be  determined.  This  may  be  done  by 
bisecting  the  angle  between  the  direction  of  a  fixed  star,  as  seen 
at^  culmination,  and  its  apparent  direction,  when  seen  at 
another  culmination  in  a  mirror  of  liquid  mercury,  called  the 
artificial  horizon.  By  a  law  of  optics,  the  apparent  depression 
below  the  horizon  equals  the  elevation  above  it,  so  that  the 
wliole  angle  equals  twice  the  altitude. 


ALTITUDE   AND   AZIMUTH   INSTRUMENT. 


31 


5  2.  The  transit  circle. — Sometimes  the  circle  of  the  transit 
instrument  is  made  of  much  larger  size  than  is  represented  in 
Fig.  11,  in  order  that  decimations  as  well  as  right  ascensions 
may  be  observed  by  it.  This  combination  of  the  transit  instru- 
ment and  mural  circle  is  called  the  transit  circle,  and  is  con- 
sidered by  some  practical  astronomers  to  possess  an  advantage 
over  the  mural  circle  in  the  steadiness  of  its  axis. 

53.  The  altitude  and  azimuth  instrument. — The  essential 
parts  of  this  instrument  are,  a  telescope  and  two  graduated 
circles,  one  vertical,  the  other  horizontal.  Fig.  16  presents  one 
of  its  more  simple  forms.  The  telescope  AB  is  movable  on  a 

Fig.  16. 


horizontal  axis  at  the  center  of  the  vertical  circle  dbc,  and  also 
on  a  vertical  axis,  passing  through  the  center  of  the  horizontal 
circle  EFG.  The  levels  g  and  />,  placed  at  right  angles  to  each 
other,  show  when  the  circle  EFGr  is  brought  to  a  horizontal 
position  by  the  tripod  screws.  The  tangent  screws,  d  and  <?, 
give  slow  motions,  one  in  a  vertical,  the  other  in  a  horizontal 
plane.  If  the  reading  of  the  vertical  circle  is  taken  when  the 


32  THE   SEXTANT. 

telescope  is  horizontal,  and  again  when  it  is  directed  to  a  star, 
the  difference  of  the  readings  is  equal  to  the  altitude  of  the  star. 
In  a  similar  manner,  if  the  horizontal  circle  is  read,  when  the 
telescope  is  directed  to  the  north,  and  read  again  when  it  is 
directed  to  a  star,  the  difference  is  its  azimuth. 

54.  The  sextant. — This  is  an  instrument  for  measuring  the 
angular  distance  between  two  points  situated  in  any  plane 
whatever.  It  is  represented  in  Fig.  17.  I  and  H  are  two 
small  mirrors,  and  T  a  small  telescope.  ID  is  a  movable  radius 
or  index,  carrying  the  index  mirror  at  the  center  of  motion,  I, 

Fig.  17. 


*r~ 


and  a  vernier  at  the  extremity,  D.  The  horizon  glass,  H,  is 
silvered  only  on  one-half  of  its  surface.  When  the  zero  of 
the  vernier  coincides  with  that  of  the  arc  at  F,  the  mirrors 
are  precisely  parallel.  If  now  we  direct  the  telescope  to  a 
star,  it  may  be  seen  in  the  transparent  part  of  the  horizon 
glass,  and  its  image  in  close  contact  with  it,  in  the  silvered  part 
This  is  owing  to  the  fact,  that  a  heavenly  body  is  so  far  dis- 
tant, that  the  rays  from  it  to  the  two  mirrors  are  sensibly  par- 
allel to  each  other. 


SPHEEICAL   PKOBLEMS.  33 

55.  To  measure  an  angle  ly  the  sextant. — Let  it  be  required 
to  measure  the  angular  distance  between  the  star  S  and  the 
moon  M.     The  telescope  being  directed  to  S,  and  the  sextant 
being  held  so  that  the  plane  of  reflection  shall  pass  through  the 
two  objects,  turn  the  index  from  F  toward  E,  until  the  image 
of  the  moon  is  brought  to  the  star,  its  nearer  limb  just  touch- 
ing S.     JSTow,  according  to  an  optical  principle,  the  angular 
distance  between  the  moon  and  its  image  is  just  twice  that  be- 
tween the  mirrors.     Therefore,  by  reading  the  vernier  at  D,  we 
obtain  the  angular  distance  between  the  star  and  the  moon's 
nearer  limb.     Again,  bring  the  further  limb  to  the  star,  and 
find  its  distance.     Half  their  sum  is  the  angular  distance  be- 
tween the  moon's  center  and  the  star. 

In  like  manner,  the  altitude  of  a  body  may  be  found,  by 
bringing  its  image  to  coincide  with  the  image  of  the  same  body 
seen  in  the  artificial  horizon.  One-half  the  angle  read  from 
the  vernier  is  the  altitude  of  the  body. 

The  graduation  on  the  limb  of  the  sextant,  for  convenience, 
corresponds,  not  to  the  actual  length  of  the  arc  passed  over  by 
the  vernier,  but  to  the  angular  motion  of  the  body,  which  is 
twice  as  rapid.  Hence,  on  the  arc  of  60°,  the  graduation 
reaches  120° ;  and  all  angles  not  greater  than  this  can  be 
measured  by  the  instrument. 

The  two  instruments  just  described  are  sometimes  conven- 
ient at  the  observatory,  but  their  chief  use  is  elsewhere.  The 
altitude  and  azimuth  instrument  is  of  great  value  in  trigono- 
metrical surveying.  The  sextant  is  important  for  the  naviga- 
tor, since  a  stationary  instrument 
cannot  be  employed  at  sea.  FiS- 18- 

56.  Spherical  problems. — 

I.  To  compute  the  sun's  right 
ascension,  declination,  or  longi- 
tude, or  the  obliquity  of  the  eclip- 
tic to  the  equator,  when  any  two 
of  the  others  are  given. 

Let  PEP'  (Fig.  18)  represent 
the  solstitial  colure,  PP'  the  axis, 
EQ  the  equator,  E'C  the  ecliptic, 


34  SPHERICAL   PROBLEMS. 

and  PSP'  a  secondary  of  the  equator  passing  through  the  sun 
S.  Then  SAE  is  the  obliquity  of  the  ecliptic,  and  ES  the  dec- 
imation of  the  sun.  And  if  its  longitude  is  less  than  90°,  AS 
is  its  longitude,  and  AE  its  right  ascension.  If  its  longitude 
is  more  than  90°,  AS  and  AE  are  the  supplements  of  longitude 
and  right  ascension.  In  both  cases  the  declination  is  north. 
When  the  sun's  place  is  represented  by  S',  and  its  longitude  is 
between  180°  and  270°,  then  the  longitude  =  180°  +  AS7,  and 
the  right  ascension  =  180°  +  AE'.  But  if  its  longitude  is 
more  than  270°,  longitude  =  360°  —  AS',  and  right  ascen- 
sion =  360°  —  AE7.  In  each  case  the  declination  is  south. 

The  triangle  AIJS  is  right-angled  at  E ;  and  by  Napier's 
rule,  any  one  of  the  parts  may  be  found,  when  twTo  others  are 
given. 

Ex.  1.  "When  the  sun's  right  ascension  is  53°  38',  and  its  dec- 
lination, 19°  15'  57",  required  its  longitude,  and  the  obliquity 
of  the  ecliptic. 

1.  Ead  .  cos  AS  =  cos  AE  .  cos  ES. 

2.  Ead  .  sin  AE  =  tan  ES  .  cot  A. 

Ans.  Long.  =  55°  57'  43".     Obi.  =  23°  27'  50J". 

Ex.  2.  On  March  3lst,  the  sun's  declination  was  observed  to 
be  4°  13'  31i",  and  the  obliquity  was  23°  27'  51" ;  required 
the  sun's  right  ascension.  Ans.  9°  47'  59". 

Ex.  3.  What  is  the  sun's  longitude  in  November,  when  its 
declination  is  21°  16'  4",  and  its  ris;ht  ascension  is  16h.  14m. 
58.4s.  \  Ans.  245°  39'  10". 

The  above  data  show  that  the  sun's  longitude  is  more  than 
180°  and  less  than  270°,  and  the  decimation  south.  The  tri- 
angle for  computation  is  AE'S'. 

Ex.  4.  The  sun's  longitude  being  8s  7°  40'  56",  and  the 
obliquity  23°  27'  42^" ;  required  right  ascension  in  time. 

Ans.  16h.  23m.  34s. 

II.  Given  the  latitude  of  a  place,  and  the  declination  of  the 
sun,  to  find  the  time  of  its  rising  and  setting. 

Let  PEP'  (Fig.  19)  be  the  meridian  of  the  place.  Z  its  zenith, 
and  HO  its  horizon.  Let  LL'  be  the  diurnal  circle  of  the  sun  ; 
ES  is  its  declination,  S  the  place  of  its  rising  and  setting,  and 
LS  the  arc  described  between  either  and  midnight.  But  LS, 
in  degrees,  equals  QE,  the  complement  of  AE.  The  angle 


SPHEEICAL   PKOBLEMS. 


35 


Fig.  19. 


S  AR  =  EAH,  which  is  measured  by 
EH,  the  co-latitude,  arid  R  is  a  right 
angle.  Therefore,  rad  .  sin  AR  = 
cot  A  .  tan  RS. 

Ex.  1.  Required  the  time  of  sun- 
rise at  latitude  52°  13'  K,  when  the 
sun's  declination  is  23°  28'  K 

We  find  AR=34°  3'  21J";  /. 
QR  =  55°  56'  38f//  =  (in  time)  3h. 
43m.  46Js.  This  is  the  time  of  sun- 
rise. The  same  subtracted  from 
12h.,  gives  8h.  16m.  13Js.  for  the  time  of  sunset. 

Ex.  2.  Required  the  time  of  sunrise  at  latitude  57°  2'  54" 
K,  when  the  sun's  decimation  is  23°  28'  N. 

Ans.  3h.  llm.  49s. 

Ex.  3.  How  long  is  the  sun  above  the  horizon  in  latitude  58° 
12'  K,  when  its  declination  is  18°  40'  S.  ? 

Ans.  7h.  35m.  52s. 

In  a  similar  manner,  if  the  decimation  of  any  heavenly  body 
be  given,  the  interval  of  time  between  its  culmination,  and  its 
rising  or  setting,  can  be  computed. 

III.  Given  the  latitude  of  a  place,  and  the  declination  of  a 
heavenly  body,  to  compute  its  altitude  and  azimuth,  when  on 
the  six  o'clock  hour-circle. 

Let  PEP'  (Fig.  20)  be  the  meridian  of  the  place,  and  P  the 
elevated  pole.  Then  PP'  rep- 
resents^  the  six  o'clock  hour- 
circle,  which  is  at  right  angles 
to  the  meridian,  and  therefore 
projected  in  a  straight  line. 
Let  the  body  cross  it  at  S,  and 
let  ZSB  be  the  vertical  circle 
passing  through  it.  In  the  tri- 
angle ASB,  AS  is  the  declina- 
tion, SB  the  altitude,  AB  the 
amplitude  or  complement  to 
the  azimuth  OB,  and  B  is  a 
right  angle. 

Ex.  1.  What  were  the  altitude  and  azimuth  of  Arcturus, 


Fig.  20. 


36 


SPHERICAL   PROBLEMS. 


when  on  the  six  o'clock  hour-circle,  latitude  51°  28'  40"  IN".,  its 
declination  being  20°  6'  50"  K  ? 

Ans.  Altitude  15°  36'  27" ;  Azimuth  77°  9'  4". 

Ex.  2.  In  latitude  62°  12'  K  the  altitude  of  the  sun  at  six 
o'clock,  A.  M.,  was  observed  to  be  18°  20'  23".  Eequired  its 
declination  and  azimuth. 

Ans.  Declination  20°  50'  12"  K  ;  Azimuth  79°  56'  4". 

IY.  Given  the  latitude  of  a  place  and  the  sun's  declination, 
to  find  the  time  A.  M.  when  it  will  cease  shining  on  the  north 
side  of  a  building,  or  the  time  P.  M.  when  it  will  begin  to  shine 
upon  it. 

Let  PEP'  (Fig.  21)  be  the  meridian  of  the  place,  ZA1ST  the 
prime  vertical,  and  S  the  place  where 
the  sun  crosses  it,  and  thus  ceases  to 
shine  on  the  north  side  of  a  vertical 
wall.  Let  PSB  be  the  hour-circle 
through  the  sun  at  S.  BS  is  the  sun's 
declination,  BAS  (=EZ)  is  the  lati- 
tude, and  AB,  changed  into  time,  will 
show  how  long  after  six  o'clock  A.  M., 
or  before  six  p.  M.,  the  sun  transits 
the  prime  vertical. 

Ex.  1.  In  latitude  42°  22'  17"  K, 
when  the  sun's  declination  is  23°  27'  36"  N.,  at  what  times  does 
the  sunshine  begin  and  end  on  the  north  and  south  sides  of  a 
building?  Ans.  7h.  53m.  38s.  A.  M.,  and  4h.  6m.  22s.  p.  M. 

Ex.  2.  How  long  does  the  sun  shine  on  the  south  side  of  a 
vertical  wall,  in  latitude  20°  30'  ]ST.,  when  the  sun's  declination 
is  20°  N  3  Ans.  Ih.  45m.  48s. 

Y.  The  latitude  and  the  sun's  declination  being  given,  to 
find  the  time  of  day  by  the  sun's  altitude. 

Let  Z  (Fig.  22)  be  the  zenith  of  the  place,  P  the  pole,  and 
S  the  place  of  the  sun.  Measure  ZS, 
the  zenith  distance  of  the  sun,  and 
correct  it  for  refraction.  PZ  is  the 
co-latitude  of  the  place,  and  PS  the 
co  declination  of  the  sun.  Therefore, 
the  sides  of  the  spherical  triangle  PZS 
are  all  known,  and  the  angle  ZPS  can 


SPHERICAL   PROBLEMS. 


37 


Fig.  23. 


be  computed ;  which,  changed  to  time,  shows  how  long  before 
or  after  noon  the  observation  was  made. 

VI.  Given  the  latitude  and  the  sun's  declination,  to  find  the 
time  when  twilight  begins  and  ends. 

The  twilight  begins  or  ends  when  the  sun  is  about  18° 
below  the  horizon  (Art.  38).  Let  Z  (Fig.  23)  be  the  zenith,  P 
the  pole,  and  S  the  place  of  the  sun  at  the  beginning  or  end  of 
twilight.  ZS  =  108°,  ZP  =  co-lat, 
PS  =  co- dec!.  The  three  sides  of 
ZPS  are  given,  to  find  the  hour-angle 
ZPS.  This  may  be  done  by  dropping 
the  perpendicular  arc  P/>,  and  using 
the  proportion  (Sph.  Trig.)  tan  J  ZS  : 
tan  i  (PS  +  ZP)  : :  tan  I  (PS  -  ZP)  : 
tan  ?  (Sp  —  Z/j).  Having  obtained 
Zp  and  S/>,  compute  the  angles  at  P, 
and  add  them  together. 

Ex.  In  lat.  42°  22',  when  does  twi- 
light begin  and  end,  at  midsummer, 
the  sun's  declination  being  23°  28'  ? 

Ans.  2h.  6m.  41s.  A.  M.  ;  9h.  53m.  19s.  P.  M. 

VII.  Given  the  right  ascension  and  decimation  of  a  body,  to 
find  its  longitude  and  latitude. 

Let  EQ  (Fig.  24)  be  the  equator,  and  P  its  north  pole,  E'C 
the  ecliptic,  and  R  its  pole,  and 
S  the  place  of  the  body.  Join 
PS  and  US,  and  draw  the  arc 
SB  perpendicular  to  PC.  PS, 
the  complement  of  declination, 
is  known  ;  likewise  HP,  which 
equals  EE',  the  obliquity.  As 
A  is  the  vernal  equinox,  SPQ 
is  the  complement  of  right  as- 
cension, and  therefore  known. 
SRC  is  the  complement  of  lon- 
gitude, and  RS  is  the  comple- 
ment of  latitude. 

In  the  right-angled  triangle  PSB,  PS  and  P  being  known, 
find  PB.     Then  RB(=RP+PB)  is  known.      Then   (Sph. 


Fig.  24. 


E 


38  THE  SUN'S   EIGHT  ASCENSION. 

Trig.)  sin  EB  :  sin  PB  : :  tan  P  :  tan  E.  Tims  E,  the  com- 
plement of  longitude,  is  found.  Then,  in  the  right-angled 
triangle  ESB,  EB  and  the  angle  E  enable  us  to  find  ES,  the 
complement  of  latitude. 

Ex.  1.  The  right  ascension  of  a  planet  was  observed  to  be 
82°  7',  and  its  declination  24°  26'  K  Calling  the  obliquity 
23°  27'  20",  what  were  the  longitude  and  latitude  of  the 
planet?  Ans.  Long.  82°  49'  30"  ;  Lat,  1°  10'  27"  K 

Ex.  2.  "What  are  the  longitude  and  latitude  of  the  star, 
whose  right  ascension  is  4h.  40m.  49s.,  and  its  declination  66° 
6'  37"  N.  ?  Ans.  Long.  79°  7'  8"  ;  Lat.  43°  24'  5"  K 


CHAPTEE  IV. 

THE     EARTH'S.  ANNUAL    MOTION    ABOUT    THE    SUN. — THE 
SEASONS. — FIGURE   OF   THE   EARTH'S   ORBIT. 

57.  Observations  of  the  sun*s  place. — If  we  employ  the  in- 
struments of  the  observatory  in  measuring  from  day  to  day  the 
right  ascension  and  declination  of  the  sun,  at  the  moment  of  its 
crossing  the  meridian,  it  will  be  discovered  that  these  quantities 
are  constantly  changing ;  or/ in  other  words,  that  the  sun  is 
constantly  shifting  its  place  in  relation  to  the  stars. 

58.  Its  right  ascension. — By  the  transit  instrument   and 
clock,  it  is  found  that  the  sun's  right  ascension  is  always  in- 
creasing by  a  quantity  which  is  not  quite  uniform,  but  which 
amounts  to  nearly  one  degree  every  day.     So  that,  in  about 
865  days,  it  describes  the  whole  360°  of  right  ascension,  and 
appears  again  in  the  same  place  among  the  stars.     This  is  the 
apparent  annual  motion  of  the  sun,  by  which  it  seems  to  pass 
round  the  heavens  from  west  to  east  once  in  a  year. 


THE   TROPICS   AND   POLAR  CIRCLES.  39 

59.  Its  decimation. — Bat  while  thus  passing  round,  it  also 
moves  alternately  north  and  south.     For,  by  measuring  the 
declination  each  day  by  the  mural  circle,  it  is  found  that  after 
passing  the   vernal   equinox,  March   21st,  its  declination    is 
north,  and  increases  to  the  summer  solstice,  June  22d,  when  it 
reaches  nearly  23|° ;  from  that  point  it  diminishes  to  zero  at 
the  autumnal  equinox,  September  22d.     The  declination  then 
becomes  south,  increasing  to  the  winter  solstice,  December  22d, 
when   it  is   23 1°,  and  thence  diminishing  to  nothing  at  the 
vernal  equinox,  on  March  21st  of  the  following  year. 

60.  The  ecliptic. — The  apparent  annual  path  of  the  sun  is 
found  by  the  foregoing  observations  to  lie  in  a  plane,  cutting 
the  celestial  sphere  in  a  circle  called  the  ecliptic  (Art.  12),  and 
inclined  to  the  plane  of  the  equator  at  an  angle  of  about  23°  27'. 
This  plane  maintains  almost  a  constant  position  among  the 
stars,  and  is  used  far  more  than  any  other  circle  of  the  sphere 
as  a  plane  of  reference. 

The  obliquity  of  the  equator  to  the  ecliptic  in  1850  was 
23°  27'  32",  and  diminishes  at  the  rate  of  48"  in  a  century. 

6 1 .  T/ie  zodiac. — This  name  is  given  to  a  zone  of  the  heav- 
ens, 16°  wide,  extending  along  the  circle  of  the  ecliptic,  8°  on 
each  side  of  it.     The  paths  of  the  principal  planets  lie  within 
this  zone.     Its  length  is  divided  into  12  signs  of  30°  each, 
having  the  same  names  and  arranged  in  the  same  order  as 
those  of  the  ecliptic  (Art.  13),  though  not  coincident  with  them. 
The  signs  of  the  zodiac  are  distinguished  from  each  other  by 
the  stars  which  occupy  them. 

62.  The  tropics  and  polar  circles. — Through  the  two  points 
of  the  ecliptic  most  distant  from  the  equator,  called  the  sol- 
stices (Art.  12),  .we  imagine  circles  to  be  drawn  parallel  to 
the  equator,  called  the  tropics.     The  northern  circle,  passing 
through  the  first  of  Cancer  on  the  ecliptic,  is  called  the  tropic 
of  Cancer ;  the  southern  one,  for  a  like  reason,  is  called  the 
tropic  of  Capricorn.    Two  other  parallels  to  the  equator,  passing 
through  the  poles  of  the  ecliptic,  and  therefore  23°  27'  from  the 
poles  of  the  equator,  are  called  the  polar  circles. 


40  ANNUAL   MOTION. 

63.  Terrestrial  zones. — On  the  terrestrial  sphere,  a  similar 
system  of  circles  divides  the  earth's  surface  into  the  well-known 
zones  of  geography,  called  the  torrid,  temperate,  mA  frigid 
zones.     The  tropics  are  the  limits  of  vertical  sunshine  in  mid- 
summer.    The  polar  circles  are  the  limits  within  which  the 
sun  makes  a  diurnal  revolution  in  midsummer  and  mid-winter, 
without  rising  or  setting. 

64.  The  annual  motion  observed  without  instruments. — If 
the  stars  were  visible  in  the  daytime,  we  should  perceive  the 
sun  making  progress  among  them  toward  the  east,  by  a  dis- 
tance equal  to  nearly  twice  its  own  breadth  every  day,  since 
the  apparent  diameter  of  the  sun  is  a  little  more  than  half  a 
degree.     But,  as  they  are  invisible  by  day,  we  detect  the  same 
fact,  when  we  notice  that  at  a  given  hour  of  the  night,  all  the 
stars  are  further  west  than  on  a  previous  night.     For  example,  at 
9  o'clock  P.  M. — that  is,  9  hours  after  noon — it  is  easily  observed 
that  there  is,  from  one  evening  to  another,  a  regular  progress 
of  all  the  stars  westward,  as  long  as  we  choose  to  watch  them. 
In  other  words,  the  sun  is  at  the  same  rate  advancing  eastward 
relatively  to  the  stars. 

Fig.  25. 


6  s.  The  annual  motion  is  a  motion  of  the  earth,  not  of  the 
sun. — There  is  abundant  evidence  that  the  motion  of  the  sun 


CHANGE   OF  SEASONS.  41 

around  the  earth,  above  described,  is  only  apparent,  and 
results  from  a  real  motion  of  the  earth  about  the  sun.  Thus, 
suppose  the  earth  to  pass  around  the  sun  S  (Fig.  25)  in  the 
orbit  ABPC,  in  the  order  of  the  signs ;  if  we  were  unconscious 
of  this  motion,  the  sun  would  appear  to  us  to  move  about  the 
earth  in  the  same  order  of  the  signs,  though,  at  any  given 
moment,  in  a  contrary  direction.  When  the  earth  is  at  B  (in 
the  sign  T,  as  seen  from  the  sun),  we  should  see  the  sun  in  the 
sign  ^  ;  when  we  reach  « ,  the  sun  is  seen  in  TTJ,  ;  and  so  on. 

66.  Cause  of  the  change  of  seasons. — The  phenomena  of  the 
seasons  are  due  to  the  fact,  that  the  two  revolutions  of  the 
earth,  one  on  its  axis,  and  the  other  around  the  sun,  are  in 

Fig.  26. 

— 
MarcJi. 


different  planes ;  in  other  words,  that  the  equator  and  the 
ecliptic  make  an  angle  with  each  other.  In  Fig.  26,  let  the 
ecliptic  be  represented  by  the  large  circle  in  the  plane  of  the 


42  HEAT  IX   SUMMER  AND   COLD   IN  WINTER. 

> 

paper.  And  suppose  the  earth  to  pass  round  the  sun  in  the 
order  of  the  signs,  T,  »,  n.  etc.,  occupying  the  position  A  on 
the  21st  of  March,  B  on  June  22d.  C  on  September  22d,  and 
D  on  December  22d. 

Next,  suppose  the  plane  of  the  equator  (represented  by  the 
straight  line  eq\  to  be  inclined  to  the  plane  of  the  paper  by  an 
angle  of  23|°,  and  always  in  the  same  direction.  The  axis  us, 
which  is  perpendicular  to  eg,  will  therefore  be  parallel  to  itself 
in  all  positions  of  the  earth.  In  the  figure,  it  is  represented  as 
everywhere  leaning  to  the  right.  At  A,  the  earth's  position 
March  21st,  the  rays  of  the  sun  just  reach  to  n  and  s  ;  so  that, 
if  the  earth  revolves  on  ns  at  that  place,  every  spot  on  its  sur- 
face will  be  one-half  the  time  in  the  light,  and  the  other  half 
in  darkness.  The  days  and  nights  are  therefore  equal.  In  this 
position,  the  plane  of  eg,  if  extended,  passes  through  the  sun — 
that  is,  the  sun  is  in  the  equator  of  the  heavens,  and  it  is  the 
time  of  the  vernal  equinox. 

In  the  position  B,  the  circle  of  illumination,  as  represented, 
reaches  beyond  n  to  the  polar  circle,  and  falls  short  of  6-  by  the 
same  distance,  the  sun  being  seen  north  of  the  equator  eq.  As  the 
earth  revolves  on  ns,  it  is  evident  that  all  places  north  of  eq  are 
longer  in  light  than  in  darkness ;  and  the  reverse  is  true  of  all 
places  south  of  eq.  It  is  now  summer  in  the  northern  hemi- 
sphere, and  winter  in  the  southern. 

At  C,  the  earth  has  reached  the  autumnal  equinox ;  the 
circle  of  illumination  passes  through  n  and  s,  and  the  phenom- 
ena are  the  same  as  at  A. 

At  D,  the  north  pole  is  turned  as  far  as  possible  into  the 
shade,  and  the  south  pole  into  the  sunlight.  The  sun  is  at  the 
tropic  of  Capricorn  ;  and  as  the  earth  rotates  on  ns,  all  places 
north  of  the  equator  experience  the  short  days  and  the  long 
nights  of  winter,  and  the  reverse  at  all  places  south  of  the 
equator. 

67.  Causes  of  heat  in  summer  and  cold  in  winter. — These 
are  two. 

\  st.  The  length  of  the  day  compared  with  the  night.  The 
heat  of  the  earth  is  passing  off  by  radiation  during  the  whole 
time,  whether  the  sun  shines  or  not.  But  the  earth  receives 


GREATEST   HEAT   AND   GREATEST   COLD. 


heat  from  the  sun,  only  while  the  sun  is  above  the  horizon. 
Hence,  the  longer  the  period  of  sunshine,  compared  with  the 
time  of  a  diurnal  revolution,  the  greater  the  heat.  For  this 
reason,  therefore,  the  summer  is  warmer  than  the  winter. 

3d.  The  different  inclination  of  the  rays  to  the  general  sur- 
face of  the  earth.  The  number  of  rays  falling  on  a  given  sur- 
•face,  varies  as  the  sine  of  inclination.  Let  AB  (Fig.  27)  be 
the  breadth  of  the  surface.  If  the  rays  fall  on  it  at  tl 
angle  ABC,  the  perpendicular 
breadth  of  the  beam  is  AC  ;  if 
at  the  angle  ABD,  the  breadth 
of  the  beam  is  AD  ;  while,  if 
they  fall  perpendicularly,  the 
breadth  of  the  beam  is  AB 
itself.  Now,  the  number  of 
rays  in  the  beam  obviously 
varies  as  its  perpendicular 
breadth.  But  these  breadths, 
AC,  AD,  and  AB,  are  as  the  sines  of  the  several  inclinations. 
In  summer,  the  sun  rises  to  a  greater  elevation  each  day  than  at 
other  seasons,  and  therefore  sheds  a  greater  quantity  of  heat  on 
that  part  of  the  earth. 

Ex.  1.  What  is  the  relative  quantity  of  direct  heat  from  the 
sun  at  noon,  on  two  equal  horizontal  areas,  one  in  latitude  75° 
K,  the  other  30P  K,  when  the  sun's  declination  is  19°  K  ? 

Ans.  As  100  :  1T5J. 

Ex.  2.  Find  the  ratio,  as  in  Ex.  1,  in  latitude  50°  N.  and 
latitude  45°  S.,  when  the  sun's  declination  is  15°  45'  S. 

Ans.  As  100  :  212.4. 

68.  Why  the  greatest  heat  is  later  than  the  summer  solstice, 
and  the  greatest  cold  later  than  the  winter  solstice. — If  the  sun 
sheds  on  a  given  surface  more  heat  each  day  than  the  surface 
loses  by  radiation,  then  the  heat  accumulates  from  day  to  day. 
This  is  the  case  during  the  long  days  of  summer ;  and  more 
heat  is  gained  than  lost,  till  a  month  or  more  after  the  summer 
solstice.  For  a  like  reason,  during  the  middle  hours  of  the  day, 
heat  is  received  from  the  sun  more  rapidly  than  it  is  lost  by 
radiation,  so  that  the  hottest  hour  is  2  or  3  o'clock  p.  M. 


44  GKEATEST  CHANGES   OF   SEASON. 

* 

In  the  winter,  on  the  contrary,  the  loss  by  radiation  exceeds 
the  quantity  received  from  the  sun,  during  all  the  shortest  days, 
so  that  the  temperature  descends  till  many  weeks  after  the 
4  winter  solstice. 

If  loss  by  radiation  were  at  a  uniform  rate  at  all  tempera- 
tures, and  the  temperature  of  successive  years  should  remain 
constant,  as  it  now  is,  then  the  greatest  heat  would  be  near  the 
autumnal  equinox,  and  the  greatest  cold  near  the  vernal  equi- 
nox, the  times  when  the  surface  receives  heat  at  the  mean  rate. 

On  the  contrary,  if  the  existing  amount  of  loss  by  radiation 
were  distributed  so  as  to  be  exactly  proportional  to  the  acces- 
sions received  from  the  sun,  there  would  be  no  change  of  tem- 
perature at  the  different  seasons  of  the  year  or  the  different 
hours  of  the  day. 

But  the  radiation  of  heat  follows  neither  of  these  laws  ;  the 
quantity  radiated  is  greater,  when  the  quantity  received  is 
greater,  but  it  does  not  vary  at  so  rapid  a  rate. 

69.  No  change  of  seasons,  if  there  were  no  obliquity. — The 
angle  between  the  planes  of  the  two  motions  of  the  earth  being 
the  cause  of  the  change  of  seasons,  it  follows  that  there  would 
be  no  such  change  if  those  motions  were  in  the  same  plane.     If, 
while  the  earth  advances  in  its  orbit  about  the  sun,  it  should 
rotate  in  the  same  direction  on  its  axis,  then  the  sun  would 
always  be  in  the  plane  of  the  equator,  and  would,  every  day  of 
the  year,  describe  the  equator  as  its  diurnal  circle,  rising  exactly 
in  the  east,  culminating  at  a  zenith  distance  equal  to  the  lati- 
tude of  the  place,  and  setting  exactly  in  the  west.     At  the 
equator,  the  sun  would  always  follow  the  prime  vertical,  and  at 
either  pole  it  would  always  be  passing  round  in  the  horizon. 

70.  The  greatest  changes  of  season,  if  the  obliquity  were 
90°. — If,  while  the  earth  revolves  on  its  axis  from  west  to  east, 
it  should  pass  around  the  sun  in  a  plane  lying  north  and  south, 
then  the  ecliptic  would  pass  through  the  north  and  south  poles, 
and  the  solstices  would  be  at  the  poles.     Hence,  at  a  station  on 
the  equator,  the  sun  would,  during  the  year,  describe  the  prime 
vertical  and  various  small  circles  parallel  to  it,  down  to  the 
north  and  south  points  of  the  horizon,  where  it  would  be 


FOKM   OF   THE   EARTH'S  ORBIT. 


stationary  alternately  at  the  times  of  the  solstices.  At  the 
equator,  therefore,  there  would  be  an  alternation  from  summer 
to  winter  every  three  months. 

At  either  pole  there  would  be  but  one  summer  and  one 
winter  in  a  year;  but  the  extremes  would  be  far  more  intense. 
For  the  sun,  in  describing  diurnal  circles  parallel  to  the 
horizon,  would  occupy  six  months  in  ascending  to  the  zenith 
and  returning  to  the  horizon,  and  the  remaining  six  months  in 
performing  corresponding  revolutions  below  the  horizon. 

At  intermediate  places,  the  extremes  of  the  seasons  would 
also  be  intermediate. 

7 1 .  Mode  of  determining  the  form  of  the  earths  orbit. — 
The  earth's  orbit  is  an  ellipse  described  about  the  sun,  which  is 
situated  in  one  of  its  foci.  This  is  ascertained  by  observing  the 
changes  in  the  sun's  apparent  diameter  throughout  the  year. 
When  the  sun  appears  smallest,  it  is  most  distant ;  and  when 
largest,  it  is  nearest.  And  its  distance,  in  all  cases,  varies  in- 
versely as  its  apparent  diameter.  Therefore,  if  the  sun's  angu- 
lar diameter  be  accurately  measured  as  frequently  as  possible, 
the  reciprocals  of  those  angles  express  the  relative  distances ; 
and  these  distances  determine  the  form  of  the  orbit. 

Thus,  suppose  the  earth  to  be  at  E  (Fig.  28),  and  that  the 
sun's  apparent  diameter  is 
measured  when  in  the  di- 
rection Ea.  After  it  has 
advanced  eastward  some 
days,  so  as  to  be  seen  in  the 
direction  E£,  let  another 
measurement  be  made  ;  and 
so  on,  at  every  opportunity 
through  the  year.  Then 
let  E«,  E5,  E<?,  etc.,  be  made 
proportional  to  the  recipro- 
cals  of  the  apparent  diame- 
ters, and  be  laid  down  at 
angles  equal  to  the  angular 
changes  of  the  sun's  place.  A  line,  a  I  m  v,  passing  through 
their  extremities,  shows  the  form  of  the  sun's  apparent  orbit 


Fig.  28. 


46  LINE   OF   APSIDES, 

about  the  earth,  and  therefore  the  form  of  the  earth's  real  orbit 
about  the  sun. 

In  this  manner,  even  while  ignorant  of  the  size  of  the  orbit, 
we  learn  that  its  form  is  an  ellipse,  and  that  the  sun  occupies 
one  of  its  foci. 

72.  Definitions  relating  to  a  planetary  orbit. — Let  E  be  the 
focus  occupied  by  the  sun,  and  am  the  major  axis  of  an  ellipti- 
cal orbit  described  about  it ;  the  nearest  point,  a,  is  called  the 
perihelion,  and  the  most  distant  point,  m^  the  aphelion.     The 
two  points  a  and  m  are  also  called  the  apsides.     The  point  a 
is  sometimes  called  the  lower  apsis,  and  m  the  higher  apsis. 
The  varying  distance,  Ea,  Ei,  E/?,  etc.,  is  called  the  radius 
vector.     If  the  major  axis,  am,  is  bisected  in  0,  the  ratio  of  EC 
to  the  semi-major  axis,  aC,  is  called  the  eccentricity  of  the 
orbit.     The   less  EC  is,  compared  with  «C,  the  less  is  the 
eccentricity,  and  the  nearer  does  the  ellipse   approach   to  a 
circle.     If  E  coincides  with  C,  the  eccentricity  is  nothing,  and 
the  orbit  is  a  circle. 

73.  The  earth's   orbit  very  nearly  circular. — The   eccen- 
tricity of  the  earth's  orbit  in  1850  was  0.01 6  77,  and  is  very 
slowly  diminishing.      This  fraction  is  about  g^, — that  is,  EC 
(Fig.  28)  is^o  of  «C.     As  «C,  in  this  figure,  is  about  one  inch 
long,  EC  should  be  only  ^  of  an  inch,  in  order  to  represent 
correctly  the  proportions  of  the  earth's  orbit.     If  it  were  thus 
drawn,  it  could  not  be  distinguished  from  a  circle  in  its  appear- 
ance ;  for  the  minor  axis,  as  may  be  easily  computed,  would  be 
shorter  than  the  major  axis  by  only  TO §00  of  an  inch. 

74.  Position  (f  the  line  of  apsides. — The  direction  of  the 
major  axis  of  the  earth's  orbit,  or  the  line  of  apsides,  is  slowly 
changing ;  but  at  present  it  passes  through  the  I  Oth  degree  of 
Cancer  and  Capricorn,  as  represented  in  Fig.  25.     The  earth 
is  at  perihelion  on  the  1  st  of  January,  and  at  aphelion  on  the 
1  st  of  July.     We  are  therefore  nearest  to  the  sun  in  the  winter 
of  the  northern  hemisphere,  and  furthest  from  it  in  the  summer. 

T5.  Distance  from  the  sun,  as  affecting  the  seasons. — The 


SIDEEEAL   TIME.  47 

intensity  of  the  sun's  heat  at  the  earth,  as  well  as  that  of  its 
light,  varies  inversely  as  the  square  of  our  distance  from  it. 
On  this  account,  the  intensity  of  heat  at  perihelion  is  to  that  at 
aphelion  as  (>12  :  592,  which  is  nearly  as  31  :  29.  Therefore, 
so  far  as  distance  is  concerned,  the  earth  receives  -^  more  heat 
on  the  1st  of  January  than  on  the  1st  of  July.  This  produces 
a  slight  effect  to  mitigate  the  severity  of  cold  in  winter  and  of 
heat  in  summer,  in  the  northern  hemisphere,  and  to  aggravate 
the  same  in  the  southern  hemisphere.  But,  on  account  of 
changes  going  on  in  the  places  of  the  equinoxes  and  apsides, 
this  modifying  effect  will  be  reversed  after  the  lapse  of  about 
10,000  years. 


CHAPTER    Y. 

SIDEREAL    TIME. — MEAN   AND    APPARENT  SOLAR  TIME. — 
THE   CALENDAR. 

76.  The  sidereal  day. — This  is  the  interval  of  time  which 
elapses  between  two  successive  culminations  of  a  star  (Art.  43). 
The  length  of  this  interval  appears  to  be  invariable,  whatever 
star  is  observed,  or  in  whatever  season  or  year  the  observation 
is  made.     On  this  account^  the  sidereal  day  is  regarded  as  the 
true  period  of  the  earth's  rotation  on  its  axis.     In  order  to 
reckon  by  sidereal  time,  the  moment  chosen  for  the  beginning 
of  each  sidereal  day  is  the  moment  when  the  vernal  equinox 
culminates.     The   sidereal   clock,   if  correct,   then   points   to 
Oh.  Om.  Os.     Each  sidereal  day  is  divided  into  24  sidereal  hours, 
each  hour  into  00  sidereal  minutes,  and  each  minute  into  60 
sidereal  seconds. 

77.  The  mean  solar  day. — This  is  the  mean  interval  between 
two  successive  culminations  of  the  sun.     It  will  be  shown  pres- 
ently, that  these  intervals  vary  throughout  the  year.     As  the 
sun,  by  the  annual  motion,  is  advancing  eastward  continually 
among  the  stars,  the  solar  day  must  always  be  longer  than  the 


48  INEQUALITY   OF  SOLAK  DATS. 

sidereal  day.  For,  if  the  sun  and  a  star  were  on  the  meridian 
of  a  place  together,  then,  while  that  place  passes  around  east- 
ward till  its  meridian  meets  the  star  again,  the  sun  has  ad- 
vanced eastward  nearly  a  degree,  and  the  place  must  revolve 
nearly  a  degree  more  than  one  revolution  before  its  meridian 
will  reach  the  sun.  This  will  require  nearly  4  minutes  of 
time;  for,  in  the  diurnal  motion,  15°  correspond  to  one  hour, 
and  therefore  1°  to  Tls  of  an  hour— that  is,  four  minutes. 

78.  The  relation  of  sidereal  time  to  mean  solar  time. — As 
the   sun,  in  its   apparent   annual   motion,  describes   360°   in 
365.24  days,  it  will,  in  one  day,  on  an  average,  pass  over  360° 
-f-  365.24  =  59'  8.35",  or  nearly  1°,  as  before  stated.     But,  by 
the  diurnal  motion,  a  given  place  on  the  earth  in  one  solar  day 
describes  360°  plus  the  above  arc.     Therefore,  360°  59'  8.35"  : 
59'  8.35"  : :  24h.  :  3m.  55.9s.  of  solar  time.     This  is  the  excess 
of  the  mean  solar  day  above  a  sidereal  day.     And  the  ratio  of 
one  sidereal  hour,  minute,  or  second  is  to  one  solar  hour,  minute, 
or  second  as  360°  :  360°  59'  8.35",— that  is,  as  1  :  1.0027379. 
Therefore,  to  reduce  a  given  period  of  time  from  the  mean  solar 
to  the  sidereal  reckoning,  multiply  by  1.0027379  ;  and  to  reduce 
sidereal  time  to  mean  solar  time,  divide  by  the  same  number. 

79.  The  apparent  solar  day. — This  is  the  actual  interval 
between  two  successive  culminations  of  the  sun.     And  this  in- 
terval changes  its  length  from  day  to  day  through  the  entire 
year,  being  sometimes  greater,  and  sometimes  less  than  the 
mean  solar  day. 

In  keeping  solar  time  by  clocks  and  watches,  it  is  customary, 
for  convenience,  to  aim  to  keep  the  mean  rather  than  the 
apparent  time,  and  to  regard  the  sun  as  going  alternately  too 
fast  and  too  slow. 

80.  First  cause  of  inequality  in  apparent  solar  days. — One 
cause  of  inequality  of  days,  as  measured  by  the  sun,  is  found 
in  the  elliptical  form  of  the  earth's  orbit,  and  the  consequent 
unequal  increments  of  longitude  made  by  the  sun  from  day  to 
day.     At  P  (Fig.  25)  the  sun  is  nearest  to  us,  and  at  A  it  is 
most  distant.     The  motion  in  the  parts  of  the  orbit  near  P 


INEQUALITY   OF  SOLAR  DAYS.  49 

would  therefore  appear  greater  than  in  the  parts  near  A,  even 
if  it  were  uniform  in  all  parts.  But,  besides  this,  as  will  be 
shown  in  Chapter  VIII ,  the  motion  is  really  greatest  at  P  and 
least  at  A.  For  both  these  reasons,  then,  the  sun,  while  in  the 
nearer  half  of  its  orbit,  passes  over  the  longest  arcs  each  day 
in  the  ecliptic — that  is  to  say,  in  longitude — and  the  shortest 
arcs,  in  the  half  most  distant  from  us.  The  sun,  in  fact,  occu- 
pies nearly  8  days  more  time  in  describing  the  remote  half 
than  the  nearer  one. 

Recollecting,  now,  that  a  solar  day  consists  of  a  sidereal  day, 
plus  the  time  of  describing  diurnally  the  arc  which  the  sun,  in 
the  mean  time,  advances  annually,  it  is  clear  that  if  this  daily 
arc  is  longer,  the  solar  day  is  longer ;  and  if  shorter,  the  solar 
day  is  shorter. 

So  far  as  this  cause  is  concerned,  therefore,  the  longest  solar 
day  would  be  the  1st  of  January,  and  the  shortest,  the  1st  of 
July  ;  and  about  half-way  from  P  to  A,  and  from  A  to  P,  the 
apparent  days  would  have  their  mean  length. 

81.  Second  cause  of  inequality  in  apparent  solar  days. — 
But  the  solar  days  are  unequal  for  another  reason — the  ob- 
liquity of  the  ecliptic  to  the  equator.  Time  is  measured  by  arcs 
of  the  equator.  But  the  sun's  daily  advance  toward  the  east 
is  made  in  the  ecliptic.  Even  if  the  daily  increments  of  the 
sun's  longitude  were  equal,  those  of  its  right  ascension  would 
be  unequal,  and  therefore  the  solar  days  unequal. 

Let  Fig.  29  represent  a  portion  of  the  celestial  sphere,  AF 
a  part  of  the  equator  projected  in  a  straight  line,  OH  a  cor- 
responding part  of  the  ecliptic,  Q  the  vernal  equinox,  S  the 
summer  solstice,  and  P  the  north  pole.  Draw  through  P  a 
few  meridians,  dividing  that  part  of  the  ecliptic  near  Q  into 
short  arcs,  to  represent  the  daily  increments  of  the  sun's  longi- 
tude on  CH,  and  of  its  right  ascension  on  AF.  These  meri- 
dians are  oblique  to  CD,  but  perpendicular  to  AB.  Hence,  as 
AQC  is  a  right-angled  triangle,  QC  is  longer  than  AQ  ;  so  also, 
DQ  is  longer  than  BQ ;  and  thus  each  part  of  CD  is  longer 
than  the  corresponding  part  of  AB  ; — that  is,  the  increments 
of  the  sun's  right  ascension,  near  the  equinoxes,  are  less  than 
those  of  its  longitude.  The  obliquity,  therefore,  by  short- 

4 


50  INEQUALITY   OF  SOLAR    DAYS. 

ening  these  increments  of  right  ascension,  shortens  the  solar 
days. 

But  if  meridians  are  drawn  to  that  part  of  the  ecliptic  near  S, 
the  arcs  GH  and  EF  are  about  parallel  to  each  other,  and  the 
increments  on  the  equator  are  not  shortened,  as  they  are  at  Q. 
But,  on  the  other  hand,  the  divergency  of  the  meridians  causes 
EF  to  be  longer  than  GH,  and  each  part  of  EF  longer  than 
the  corresponding  part  of  GH.  At  the  solstices,  therefore,  the 
increments  of  right  ascension  are  lengthened  by  the  divergency 
of  the  meridians,  and  hence  the  solar  days  are  lengthened  also. 
About  midway  between  the  equinox  and  solstice,  the  two 
effects  just  described  neutralize  each  other,  and  the  daily  arcs 
of  right  ascension,  so  far  as  this  cause  is  concerned,  are  at  their 
mean  value. 

Fig.  29. 


82.  Location  of  extreme  and  mean  solar  days  from  each 
cause. — Suppose  the  first  cause  alone  in  operation,  and  that  the 
sun  and  a  uniform  clock  agreed  with  each  other  at  F  (Fig.  25), 
on  the  1st  of  January.  Then,  as  the  solar  days  are  longer  than 
their  mean,  the  sun  becomes  slower,  compared  with  the  clock, 
from  day  to  day,  for  about  three  months,  when  the  days  will 
have  reached  their  mean  length,  at  a  point  near  half-way  from 
P  to  A.  Afterward,  the  days  being  diminished  below  the 
mean,  the  sun  slowly  gains  on  the  clock,  and  catches  up  with  it 
at  A,  July  1  st.  But  the  days  now  being  shortest  of  all,  the  sun 
is  immediately  in  advance  of  the  clock,  and  most  of  all  at  a 
point  half-way  from  A  to  P.  The  gain  and  loss  compensate 


EQUATION  OF  TIME.  51 

each  other  from  A  to  P,  as  they  did  from  P  to  A.  Thus  mean 
and  apparent  time  would  agree  twice  in  a  year,  at  intervals  of 
six  months,  if  eccentricity  of  orbit  were  the  only  cause  of 
irregularity. 

Again,  if  the  second  cause  alone  existed,  and  we  suppose  the 
sun  and  clock  to  agree  at  the  equinox  Q  (Fig.  29),  then  the  sun 
gains  on  the  clock  every  day,  on  account  of  the  short  arcs  of 
right  ascension  near  Q.  In  about  1£  months,  however,  the 
days  reach  their  mean  length,  the  sun  begins  to  lose  what  it 
has  gained,  and  at  S,  June  22d,  the  sun  and  clock  are  again  to- 
gether. But  the  sun  is  now  losing,  falls  behind  the  clock,  and 
is  furthest  behind  midway  between  the  solstice  and  the  next 
equinox.  The  autumnal  equinox  and  the  winter  solstice  are,  in 
like  manner,  points  of  time  at  which  the  clock  and  sun  agree 
with  each  other.  Thus,  if  the  second  were  the  only  cause  of 
irregularity,  the  mean  and  apparent  time  would  agree  four 
times  in  a  year,  at  intervals  of  about  three  months  each. 

83.  The  equation  of  time. — The  difference  between  mean 
time  and  apparent  time,  on  any  given  day,  is  the  equation  of 
time  for  that  day.     If  the  sun  is  slow,  the  equation  must  be 
added  to  the  apparent  time ;  if  fast,  it  must  be  subtracted  from 
it,  in  order  to  give  mean  time. 

"We  have  seen  by  the  two  preceding  articles  that,  on  account 
of  eccentricity  of  orbit,  the  equation  would  be  reduced  to  zero 
twice  in  a  year ;  and,  on  account  of  obliquity  of  ecliptic  and 
equator,  it  would  be  zero  four  times  in  a  year.  The  joint  effect 
of  these  two  causes  is,  to  reduce  the  equation  to  zero  four  times 
in  a  year,  at  unequal  intervals  of  time. 

84.  The  equation  of  time  represented  graphically. — The 
ordinates  of  the  curves  in  Fig.  30  exhibit  to  the  eye  the  equation 
of  time  as  depending  on  each  cause  by  itself,  and  on  the  two 
conjointly.     The  relative  lengths  of  the  ordinates  above  and 
below  AB  show  the  positive  and  negative  equations,  as  caused 
by  eccentricity,  and  those  on  CD  the  equations  as  caused  by 
obliquity ;  while  the  algebraic  sum  of  these  on  each  vertical 
line,  gives  the  resultant  effect  on  the  line  EF.     The  figure 
shows   that   the   equation   reaches   its  first  maximum,    +   14: 
minutes,  on  the  llth  of  February;  its  first  minimum,  —  4 


52 


THE   JULIAN   CALENDAR. 


minutes,  May  14th ;  its  second  maximum,  +  6  minutes,  July 
26th ;  and  its  second  minimum,  —  U>  minutes,  November  2U. 
The  four  times  of  agreement,  when  the  equation  is  zero,  are 
?hown  by  the  intersections  ;  they  occur  April  15th,  June  15th, 
September  1st,  and  December  22d. 


Fig.  30. 
|  Jan.  |  Feb.  |  Mar.      April.    May.  |  June.  |  July.  )  Aug.     Sept  j    Oct 


Xov.  |  Dec.    1 


A 


^F 


-4 


85.  Civil  and  astronomical  time. — The  mean  solar  day,  when 
employed  for  civil  purposes,  is  supposed  to  begin  and  end  at  mid- 
night, and  is  divided  into  hours,  numbering  from  1  to  12  A.  M., 
and  then  from  1  to  12  p.  M.     But  the  astronomical  day  (which  is 
also  the  mean  solar  day)  begins  and  ends  at  noon,  12  hours 
later  than  the  corresponding  civil  day,  and  its  hours  are  counted 
from  1  to  24.     Thus,  the  astronomical  date,  April  12d.  20h.,  is 
the  same  as  the  civil  date,  April  13th,  8  o'clock  A.  M. 

86.  The  Julian  calendar. — The  period  in  which  the  sun 
passes  from  the  vernal  equinox  to  the  same  point  again,  is 
called   the   tropical  year.     In   that  period  the  round  of  the 
seasons  is  exactly  completed.     The  length  of  the  tropical  year 
is  365d.  oh.  48m.  46.15s.     This  is  so  near  365i  days,  that  in 
the  adjustment  of  the  calendar  made  by  Julius  Caspar  '(hence 
called  the  Julian  calendar),  three  successive  years  were  made 
to  contain  365  days  each,  and  the  fourth  366  days.     The  addi- 
tional day  is  called  the  intercalary  day.     In  this  calendar  it 
was  introduced  by  reckoning  twice  the  6th  day  before  the 


THE  GREGORIAN  CALENDAR.  5.3 

Kalends  of  March ;  and  hence  the  year  containing  this  addi- 
tional day  was  called  the  bissextile.  The  intercalary  day  is 
now  the  29th  of  February,  and  the  year  containing  such  a  day 
is  called  leap-year. 

87.  The  Gregorian  calendar. — By  calling  the  tropical  year 
365^  days,  the  Julian  calendar  makes  it  more  than  11  minutes 
too  long,  and  the  intercalation  of  one  day  in  four  years  is 
therefore  too  great.     This  excess  amounts  to  more  than   18 
hours  in  a  century.     Hence,  by  dropping  the  intercalary  day 
three  times  in  four  centuries,  the  adjustment  is  nearly  complete. 
The  Julian  calendar,  thus  amended,  is  called  the  Gregorian 
calendar,  because  adopted  under  Pope  Gregory  XIII.     At  that 
time,  1582,  the  vernal  equinox,  by  the  error  of  the  Julian  cal- 
endar, had  fallen  back  to  March  llth.     To  bring  the  equinox 
to  its  proper  date,  10  days  were  first  dropped,  (the  5th  being 
called  the  15th),  and  then  the  following  system  was  adopted. 

Every  year,  not  exactly  divisible  by  4,  has  365  days. 

Every  year,  divisible  by  4,  and  not  by  100,  has  366  days. 

Every  year,  divisible  by  100,  and  not  by  400,  has  365  days. 

Every  year,  divisible  by  400,  has  366  days. 

The  Gregorian  calendar  will  not  be  correct  perpetually,  but 
the  error  will  not  amount  to  a  day  in  4,000  years. 

The  nation  of  Russia  has  not  yet  adopted  the  Gregorian  cal- 
endar, so  that  there  is  now  a  discrepancy  of  12  days  between 
their  dates  and  those  of  other  nations.  The  reckoning  still 
used  by  them  is  known  as  old  style,  and  is  distinguished  by 
appending  the  letters  O.  S.  to  every  date. 

88.  How  to  compare  days  of  the  month  and  of  the  week  in 
passing  from  one  year  to  another. — A  common  year  of  365 
days  contains  52  weeks  and  one  day ;  a  leap-year  contains  52 
weeks  and  two  days.     Hence,  a  year  usually  begins  a  day  later 
in  the  week  than  the  year  previous.     And,  generally,  any  day 
of  any  month  is  one  day  later  in  the  week  than  the  same  day  of 
the  preceding  year.     Thus,  July  4th,  1865,  falls  on  Tuesday; 
1866,  on  Wednesday;  1867,  on  Thursday.     But,  in  leap-year, 
this  rule  applies  only  till  the  end  of  February.      From  that 
time  to  the  same  date  in  the  year  following,  every  day  of  a 


54  CENTRIFUGAL   FORCE. 

month  falls  two  days  later  in  the  week  than  in  the  previous 
year.  Thus,  July  4th,  1871,  is  Tuesday;  1872,  Thursday. 
And  February  2d,  1872,  is  Friday ;  1873,  it  is  Sunday. 

Table  I.,  at  the  -end  of  the  volume,  contains  a  complete  cal- 
endar for  77  centuries. 


CHAPTEE  VI. 

CURVILINEAR  MOTION. — SPHEROIDAL  FORM  OF  THE   EARTH.— 
ITS  DENSITY. — PROOFS  OF  ITS  ROTATION   ON  AN  AXIS. 

89.  Projectile  and  centripetal  forces. — Motion  in  a  curve 
line  is  always  the  effect  of  two  forces ;  one,  an  vinpulxe  which, 
acting  alone,  would  have  caused  a  uniform  motion  in  a  straight 
line,   and   whose  influence  is  always  retained  in   the   curve- 
motion  ;  the  other,  a  continued  force,  which  constantly  urges 
the  moving  body  toward  some  point  out  of  the  original  line  of 
motion.     The  first  is  called  the  projectile  force,  the  other  the 
centripetal  force.     If  the  action  of  the  latter  were  to  cease  at 
any  moment,  the  body  by  its  inertia  would  from  that  moment 
continue  uniformly  in  the  direction  in  which  it  was  then  mov- 
ing.    Such  motion  in  the  tangent  may  be  regarded  as  the 
effect  of  an  impulse  first  given  in  the  direction  of  that  tangent. 
This  supposed  impulse  is  the  projectile  force  for  the  moment  in 
question ;   but  it  is  in  truth  the  resultant  of  the  original  im- 
pulse, and  the  infinite  series  of  actions  already  produced  by  the 
centripetal  force. 

The  centripetal  force  is  infinitely  small  compared  with  the 
projectile  force.  For,  if  not,  the  curve  would  depart  by  a  finite 
angle  from  the  tangent ;  whereas,  by  the  very  nature  of  the 
relation  of  a  curve  to  its  tangent,  the  angle  is  infinitely  small : 
therefore  the  deflecting  force  is  infinitely  small.  But  it  pro- 
duces finite  deflection  after  a  time,  because  its  action  is  inces- 
santly repeated. 

90.  Centrifugal  force. — "When   a  body  moves   in  a  curve, 


CENTIUFUGAL   FORCE.  55 

since  by  its  inertia  it  tends  to  proceed  in  the  tangent  at  that 
point,  there  is  a  continual  outward  pressure  directed  from  the 
center  of  force :  this  is  called  the  centrifugal  force.  It  is 
always  opposed  to  the  centripetal  force,  and  in  circular  motion 
is  always  equal  to  it.  It  must  not  be  viewed  as  a  third  force 
introduced  to  explain  curvilinear  motion,  but  as  that  infinitesi- 
mal component  of  the  projectile  force  which  acts  in  opposition 
to  the  centripetal  force. 

9 1 .  First  law  of  centrifugal  force  in  circular  motion. — When 
a  body  moves  in  a  circular  path,  its  centrifugal  (or  centripetal) 
force  varies  as  the  square  of  the  velocity  divided  by  the  radius. 
Let  Ab  (Fig  31)  =  v,  the  space  passed  over  in  one  second. 
The  projectile  force  is  then  represented  by  AB,  and  the  body 
would  move  in  that  line  uniformly,  were  it 

not  for  the  centripetal  force  A#,  acting  to- 
ward E,  and  thus  deflecting  it  into  Ab.    Call 
the  centripetal  force  <?,  and  the  radius  o 
circle  r.     Now  (as  Ab  may  represent 
the  arc  or  its  chord)  Aa  :  Ab  : :  Ab  :  AD ; 

or,  c :  v  : :  v  :  %r ;  .•.<?=—,  which  varies  as  — . 

As  the  centripetal  and  centrifugal  forces  are 
equal  in  circular  motion,  c  may  represent  either  in  value,  though 
they  are  opposite  in  direction.  Hence,  in  a  given  circle,  where 
r  is  constant,  the  force  either  toward  or  from  the  center  varies 
as  -y2,  the  square  of  the  velocity.  In  whirling  a  ball,  for  in- 
stance, with  a  string  of  given  length,  if  the  velocity  is  doubled, 
the  strain  upon  the  string  (the  centrifugal  force)  is  four  times 
as  great,  and  the  strength  of  the  string  (the  centripetal  force) 
needs  also  to  be  four  times  as  great.  So,  if  a  train  of  cars  goes 
round  a  curve  with  a  velocity  1J  times  that  which  is  intended, 
its  tendency  to  be  thrown  from  the  track  is  increased  2£  times. 

92.  Second  law  of  centrifugal  force  in  circular  motion. — 
When  the  path  of  a  body  is  circular,  its  centripetal  or  centrif- 
ugal force  varies  as  the  radius  of  the  circle  divided  by  the. 
square  of  the  time  of  revolution. 

Let  t  =  the  time  of  describing  the  whole  circumference 


56 


LOSS  OF   WEIGHT. 


and  let  the  velocity  per  second  —  v.     Therefore,  Z^r  =  vt,  and 
,.  „<  =  *'    But  (Art.  91)  .  =       =  *,  which 


varies  as  -,. 
t 

Hence,  if  the  time  of  revolution  is  the  same,  the  attraction  to 
the  center  must  be  increased  as  the  radius  is  increased  ;  for  then 
c  oo  T.  Thus,  if  a  string  is  twice  as  long,  it  must  have  twice 
the  strength,  in  order  to  whirl  a  ball  at  the  same  rate  of 
revolution. 

93.  Centrifugal  force  on  the  earth's  surface.  —  As  the  earth 
makes  its  diurnal  rotation,  all  free  particles  upon  it  are  in- 
fluenced by  the  centrifugal  force.     Let  NS  (Fig  32)  be  the 
axis,  and  A  a  particle  describing  a  circle  with  the  radius  AO. 
If  AB,  in   the  plane   of  that 

circle,  represent  the  centrifugal 

force,  resolve  it   into  AD   on 

CA    produced,  and  AF,  tan- 

gent  to  the    meridian    NQS. 

The  effect  of  AD  is  to  dimin- 

ish the  weight  of  the  particle, 

while  the  effect  of  AF  is  to 

urge  it  horizontally  toward  the 

equator.     If  the  surface,  then, 

consists  of  yielding  matter,  as 

water,  the  spherical  form  can 

not  be  retained,  but  the  parts  about  the  poles,  ]ST  and  S,  will  be 

depressed,  and  those  about  the  equator,  EQ,  will  be  elevated. 

At  each  point  between  the  pole  and  the  equator,  a  particle  is 

held  in  equilibrio,  by  that  component,  AF,  of  the  centrifugal 

force  which  urges  it  toward  the  equator,  and  that  component 

of  gravity  which  urges  it  down  the  inclined  surface  toward  the 

pole. 

94.  Loss  of  weight  at  the  equator  caused  by  rotation.  —  Let 
the  weight  of  a  body,  w,  be  taken  to  express  the  force  of 
gravity,  and  let  \g  (=  16  f?  feet)  be  the  distance  fallen  through 
by  this  body  in  one  second.     Now,  c  is  the  force  by  which  Aa 


LOSS  OF  WEIGHT  AT  EQUATOR.  57 

(Fig.  31)  is  described  in  one  second ;  and  A.a  (used  as  the 

27T2/' 

measure  of  c)  =  — 75—  (Art.  92).     Hence, 


w  :  c::  \g\ 


Using  the  values  of  the  letters  in  the  fraction,  we  obtain  c,  the 
centrifugal  force,  in  terms  of  w,  the  weight  of  the  body. 

The  equatorial  radius  of  the  earth,  r,  is  3962.8  miles  = 
20,923,584  feet. 

The  earth  makes  one  rotation  in  24  sidereal  hours  =  86,400 
sidereal  seconds.  Reducing  this  to  solar  seconds  (Art.  78),  we 
find 

t  =  86,164s.     Hence, 


x  20,923,584         w_ 
~~  " 


321  x  8^164a~~  "  289* 


And,  since  the  centrifugal  force  at  the  equator  acts  directly 
from  the  center,  a  body  at  the  equator  loses  -yjff  of  its  weight 
by  the  rotation  of  the  earth. 

95.  Loss  of  weight  by  rotation  at  other  latitudes.  —  Since  c 
varies  as  r  (Art.  92),  the  centrifugal  force  is  greatest  at  the 
equator,  and  zero  at  the  poles,  and  the  force  at  the  equator 
is  to  that  at  any  latitude  A  (Fig.  31)  as  QC  :  AO  —  that  is,  as 
rad  :  cos  lat.     But,  except  at  the  equator,  the  centrifugal  force 
does  not  directly  oppose  gravity.     If  AB  is  the  whole  centrif- 
ugal force  at  A,  AD  is  the  component  of  it  which  acts  against 
gravity.     But  AB  :  AD  :  :  AC  :  AO  :  :  rad  :  cos  lat.     So  that 
the  loss  of  weight  is  diminished  again  in  the  same  ratio  as  be- 
fore.    Therefore,  the  loss  of  weight  at  the  equator  is  to  that  at 
any  given  latitude,  as  rad  :  cos2  of  latitude. 

96.  Whole  loss  of  weight  at  the  equator.  —  It  is  found  by 
observations  made  with  the  pendulum,  that  the  weight  of  a 
body  at  the  equator  is  T^T  less  than  that  at  the  poles.     But  the 


58 


SPHEROIDAL   FORM   OF   THE   EARTH, 


loss  from  centrifugal  force  is  only  ^ig-  Subtracting  this  from 
Ti7,  the  remainder  is  very  nearly  T J6>  a  l°ss  °f  weight  at  the 
equator  which  must  be  ascribed  to  some  other  cause.  This 
cause  is  the  oblateness  itself,  by  which  the  equator  is  more 
distant  from  the  center  than  the  poles  are. 

97.  Spheroidal  form  of  the  earth  found  ~by  measurement. — 
Not  only  is  the  oblate  form  of  the  earth  inferred  from  its  rota- 
tion on  its  axis,  but  the  measurement  of  the  length  of  a  degree 
of  latitude,  at  various  distances  from  the  equator,  proves  that 
the  meridians  of  the  earth  are  ellipses,  whose  major  axes  are  in 
the  plane  of  the  equator,  and  their  common  minor  axis  a  line 
joining  the  poles.  If  the  meridians  were  circles,  all  the  degrees 
of  latitude  would  be  of  the  same  absolute  length,  but  it  has 
been  ascertained,  by  numerous  and  most  accurate  trigometri- 
cal  surveys,  that  the  length  of  a  degree  of  latitude  is  least  at 
the  equator,  and  increases  toward  the  poles.  But  if  the  degree 
lengthens  as  we  go  toward  the  pole,  then  the  radius  must 
lengthen  in  the  same  proportion,  and  therefore  the  curve,  belong- 
ing to  a  larger  circle,  must  become  more  flattened.  And  this 
change  of  curvature  belongs  to  an  ellipse,  not  to  a  circle.  Thus, 
at  Q  (Fig.  33)  the  degree  is  shortest,  longer  at  K,  still  longer  at 
L,  and  so  on  to  the  pole. 
The  center  of  the  arc  Q 
is  at  A,  nearer  than  the 
center  of  the  earth ;  the 
center  of  K  is  B,  of  L  is 
D,  and  of  the  polar  arc 
it  is  F,  beyond  the  cen- 
ter C.  Thus,  the  cen- 
ters of  curvature  of  the 
elliptical  quadrant  QN 
lie  on  the  curve  ABDF, 
which  is  the  evolute  of 
that  quadrant.  Each 
meridian  quadrant  is  in 
like  manner  the  involute  of  a  curve,  and  their  four  evolutes 
form  the  figure  AFGH  about  the  center.  No  part  of  a  meri- 
dian has  its  center  of  curvature  at  the  center  of  the  earth. 


MASS   OF   THE   EAKTH. 


59 


The  following  numbers  express  both  the  size  and  the  form  of 
the  earth : 


Equatorial  diameter 
Polar  diameter 
Mean  diameter 
Difference  of  diameters 


7925.004  miles. 
7899.110     " 
7912.352     « 
26.494     " 


The  difference  of  diameters  is  -^  of  the  mean  diameter; 
this  is  called  the  compression  of  the  poles,  or  the  ellipticity  of 
the  earth. 

So  slight  is  the  oblateness  above  described,  that  an  exact 
model  of  the  earth  could  not  be  distinguished  by  sight  or  touch 
from  a  perfect  sphere. 

The  volume  of  the  earth  =  (7912.352)3  x  g  =  259,400,000,000 
cubic  miles. 


98.  The  equatorial  belt. — If  we  imagine  a  sphere  con- 
structed on  the  polar  diameter  of  the  earth,  the  difference  be- 
tween the  sphere  and  spheroid  will  be  a  sort  of  shell  or  ring, 
thirteen  miles  thick  at  the  equator,  and  growing  thinner  on 
every  side  to  the  poles.  This  is  sometimes  called  the  equa- 
torial ring  or  belt  of  the  earth,  and  it  produces  sensible  effects 
on  the  earth's  relations  to  the  moon  and  sun. 


99.  Weight  and  density  of 
the  earth. — The  earth's  mass, 
and  therefore  its  density,  can 
be  obtained  by  comparing  the 
effects  produced  upon  a  plumb- 
line,  by  the  earth  and  a  moun- 
tain of  known  weight.  Let  M 
(Fig.  34)  be  an  abrupt  moun- 
tain situated  alone  on  a  plain, 
and  let  a  station,  33,  be  selected 
on  the  north  side  of  it,  and  an- 
other, D,  in  the  same  meridian; 
on  its  south  side,  for  measuring 
the  zenith  distances  of  stars.  If 


Fig.  34. 


60  DIUKNAL   KOTATION. 

the  mountain  were  not  present,  the  plumb-line  of  the  zenith 
sector  would  hang  in  the  lines  B  and  D,  and  would  mark  E 
and  G  as  the  zeniths  of  the  stations.  But  the  attraction  of 
the  mountain  draws  the  plumb-line  toward  it,  so  as  to  point  to 
the  false  zeniths  E'  and  G'.  When  the  star  S,  therefore, 
culminates,  its  apparent  zenith  distance,  SE',  is  measured  at 
one  station,  and  at  another  culmination,  SG'  is  measured.  The 
difference,  SE'  —  SG',  is  the  distance  between  the  apparent 
zeniths.  The  distance,  EG,  between  the  true  zeniths,  is  the 
same  as  the  difference  of  latitude  between  the  stations  B  and 
D.  Let  a  trigonometrical  survey,  therefore,  be  made  around 
the  mountain,  and  thus  the  arc  BD,  or  its  equal  EG,  be  found. 
E'G'  —  EG  —  the  sum  of  the  two  angles  by  which  the  plumb- 
line  is  drawn  from  a  vertical  position  at  the  two  stations. 
The  volume  and  density  of  the  mountain  being  measured,  and 
the  angle  being  found,  as  above,  by  which  it  draws  a  plumb- 
line  from  a  true  vertical,  we  have  the  means  of  determining 
the  mass  of  the  earth.  And,  as  its  volume  is  known,  its 
density  is  inferred.  Observations  of  this  kind  were  made  near 
Mount  Schehallien,  Scotland,  by  Dr.  Maskelyne,  who  found 
the  deviation  of  the  plumb-line  to  be  a  little  more  than  6". 

The  mean  density  of  the  earth,  as  deduced  from  a  great 
number  of  results,  obtained  by  this  and  other  methods,  is  5.67, — 
that  is,  the  earth,  as  a  whole,  is  5.67  times  the  weight  of  the 
same  volume  of  water.  Calling  the  weight  of  a  cubic  foot  of 
water  62£  Ibs.,  the  weight  of  the.  earth  is  somewhat  more  than 
6,000,000,000,000,000,000,000  tons. 

1 OO.  Proofs  of  the  earth? s  diurnal  rotation. 

1.  To  suppose  the  earth  to  rotate  eastward  on  its  axis,  is  the 
only  reasonable  way  of  explaining  the  fact,  that  all  the  mill- 
ions of  fixed  stars,  at  various  and  immense  distances  from  us, 
in  large  and  in  small  circles  of  the  sphere,  perform  their  ap- 
parent revolutions  about  us  in  precisely  the  same  length  of  time 
— viz.,  one  sidereal  day. 

2.  Without  supposing  the  earth  to  rotate  on  its  axis,  we  can 
not  account  for  the  oblate  form,  of  the  waters  of  the  ocean. 
Whatever  form  the  solid  parts  might  have,  the  movable  portion 
would  be  spherical,  if  the  earth  were  at  rest.     Moreover,  the 


KOTATION   OF  THE   EAETH.  61 

degree  of  oblateness  is  exactly  that  which  is  required  on  a 
sphere  having  the  diameter  and  mass  of  the  earth,  if  it  be  sup- 
posed to  rotate  once  in  24  hours. 

3.  The  weight  of  a  body  at  the  equator,  compared  with  that 
at  the  poles,  is  too  small  to  be  wholly  accounted  for  by  in- 
creased distance.     Centrifugal  force,  arising  from  rotation,  can 
alone  explain  the  remaining  difference. 

4.  A  body  dropped  from  a  great  height  strikes  further  east 
than  the  vertical  line  in  which  it  began  to  fall.     If  the  earth 
rotates,  the  top  of  a  tower  moves  faster  than  the  base ;  and 
therefore   a  body  let   fall  from   the  top,  retaining  the  east- 
ward motion  of  that  point,  will  strike  further  east  than  the 
base.     At  the  equator,  this  distance  would  be  near  2  inches, 
for  a  fall  of  500  feet.     Numerous  experiments  on  the  fall  of 
bodies  through  great  distances  have  been  very  carefully  made 
by  different  individuals,  and  in  different  latitudes.     And  they 
all  concur  in  proving  that  a  body  in  falling  deviates  from  a 
vertical  line  toward  the  east. 

5.  It  is  proved  by  the  vibrations  of  a  pendulum  that  the 
earth  rotates  eastward.     Let  us  suppose  a  weight  to  be  sus- 
pended by  a  long  fine  wire,  and  then  made  to  vibrate  in  a 
plane.     The  plane  in  which  the  wire  and  weight  move  is  ver- 
tical, and  passes  through  the  point  of  suspension.    .The  weight 
itself  may  be  considered  as  describing  a  straight  horizontal  line. 
On  account  of  inertia,  the  weight  tends  to  keep  always  in  the 
same  line,  or  (if  the  point  of  suspension  be  moved)  in  a  line 
parallel  to  itself.     And  it  will  always  remain  strictly  parallel  to 
itself,  so  long  as  it  can  at  the  same  time  remain  horizontal,  and 
in  a  vertical  plane  passing  through  the  point  of  suspension. 

Thus,  if  at  the  equator  the  weight  be  made  to  vibrate  north 
and  south — that  is,  in  the  plane  of  a  meridian— it  will  continue 
to  do  so  without  deviation,  as  the  earth  rotates  eastward,  be- 
cause it  will  thus  remain  moving  horizontally  in  a  plane  which 
passes  through  the  point  of  suspension,  though  that  plane  is 
continually  changing.  In  this  case,  the  lines  in  which  the 
weight  vibrates  are  all  parallel  among  themselves. 

If  the  experiment  be  tried  at  the  pole,  and  the  weight  be 
made  to  vibrate  in  the  plane  of  a  certain  meridian,  the  point  of 
suspension  does  not  move  from  its  place,  but  only  revolves  in 


62  FOKM   OF   THE   SUN. 

it ;  and  while  the  earth  revolves  15°  per  hour,  the  weight  pre- 
serving its  own  plane  of  vibration,  will  s?em  to  shift  that  plane 
15°  per  hour  in  the  contrary  direction,  keeping  pace  with  the 
stars  in  their  diurnal  motion. 

At  localities  between  the  equator  and  the  pole,  the  line  of 
vibration  remaining  horizontal,  and  in  a  vertical  plane  which 
passes  through  the  point  of  suspension,  can  not  at  the  same  time 
preserve  its  parallelism.  But  it  will  come  as  near  fulfilling 
this  condition  as  possible.  Its  north  extremity  will  deviate 
eastward  from  the  meridian  more  or  less,  according  as  it  is 
nearer  the  pole  or  the  equator.  It  is  proved  that  the  deviation 
per  hour  is  to  15°  as  the  sine  of  latitude  to  radius. 

"When  experiments  are  performed  with  sufficient  care,  it  is 
found  that  the  pendulum  actually  deviates  eastward  from  the 
meridian,  and  at  a  rate  corresponding  well  with  the  calculated 
result.  The  pendulum  thus  furnishes  evidence  that  the  earth 
rotates  on  its  axis. 

The  above  is  known  as  Foucault's  experiment. 

6.  It  will  be  seen  hereafter  that  the  motion  of  the  equinoc- 
tial points  toward  the  west,  called  the  precession  of  the  equi- 
noxes, affords  an  independent  proof  of  the  earth's  diurnal 
motion. 


CHAPTEK  VII. 

THE   SUN. — SOLAR  SPOTS. — CONDITION   OF  THE  SUN'S 
SURFACE. — THE   ZODIACAL  LIGHT. 

1O1.  The  form  of  the  sun. — The  disk  of  the  sun  is  always 
circular.  And,  as  it  presents  all  sides  toward  us  in  its  rotation, 
we  infer  that  its  form  must  be  spherical.  But  since  it  rotates 
on  an  axis,  and  its  surface  is  in  a  fluid  state,  it  might  be  ex- 
pected to  reveal  a  spheroidal  form.  The  reasons  why  it  does 
not  are,  that  the  force  of  gravity  on  the  sun  is  very  great,  and, 
in  consequence  of  the  slowness  of  its  rotation,  the  centrifugal 
force  is  small.  It  appears  by  calculation  that  the  angle  sub- 
tended by  the  equatorial  and  the  polar  diameters  can  not  diffei 


DIMENSIONS   OF   THE   SUN.  63 

from  each  other,  except  by  a  small  fraction  of  a  second.     Its 
oblateness  is,  therefore,  too  slight  to  be  perceived. 

102.  Distance  of  the  sun,  and  size  of  the  earth's  orbit. — 
The  sun's  horizontal  parallax  is  8." 5116.  Therefore,  the  distance 
of  the  sun  from  the  earth  is  found  (Fig.  4)  by  the  proportion, 

sin  8."5776  :  rad  ::  3956.176  :  95,134,000;         ^: 

which  is  the  distance  in  miles  from  the  earth  to  the  sun. 

The  circumference  of  the  earth's  orbit,  or  the  distance  trav- 
eled by  the  earth  each  year,  is  r  ^  — 
05,134,000  x  27T  =?  584,000,000  miles. 

103.  Velocity  of  the  earth  on  its  axis  and  in  its  orbit  com- 
pared.— In  the  diurnal  motion,  a  place  on  the  equator  describes 
nearly  25,000  miles  in   24   hours — that  is,  more  than  1,000 
miles  per  hour,  or  about  17  miles  in  a  minute.     In  the  annual 
motion,  the  earth  describes  near  600,000,000  miles  in  365£ 
days,  thus  passing  over  a  distance  of  1,600,000  miles  each  day  ; 
which  is  about  1,140  miles  in  a  minute,  or  19  miles  in  a  second. 
The  earth's  velocity  in  its  orbit  is  about  67  times  as  great  as 
that  of  the  equator  in  the  diurnal  motion. 

104.  To  find  the  dimensions  of  the  sun. — The  angle  sub- 
tended by  the  sun's  diameter  may  be  measured  by  instruments. 
Let  AES  (Fig.  35)  equal  one  half  the  measured  angle.     Then 
we  have  rad  :  sin  AES  : :  ES  :  AS,  the  semi-diameter  of  the 
sun.     As  the  sun's  mean  apparent  semi-diameter  is  16'  1".5, 
and  ES  is  95,134,000   miles,  we  find  the   sun's  radius  near 
443,500,  and  therefore  its  diameter  887,000  miles. 

Fig.  35. 


The  sun's  diameter  is  about  112  times  that  of  the  earth. 
And,  since  spheres  vary  as  the  cubes  of  their  diameters,  the 
volume  of  the  sun  to  that  of  the  earth  is  as 

11 23  :  I3  ::  1,400,000  :  1.  nearly. 


DIUKN'AL  EOTATION  OF  THE  SUN. 


1O5.  The  sun's  mass  and  density. — It  is  found,  by  methods 
to  be  described  hereafter,  that  the  sun  does  not  exceed  the 
earth  in  mass  nearly  so  much  as  it  does  in  volwne.  While 
the  volumes  are  as  1,400,000  :  1,  the  masses  are  about  as 
354,000  :  1. 

The  density  of  the  sun,  therefore,  is  to  that  of  the  earth  as 
354,000  :  1,400,000  ::  1  :  4,  nearly. 


1O6.  Force  of  gravity  at  the  surface  of  the  sun.  —  When 
the  relative  masses  and  diameters  of  bodies  are  known,  it  is 
easy  to  find  the  relative  force  of  gravity  on  their  surfaces.  For 

G  QO  -yy2  (Nat.  Phil.,  Art.  15),  where  G  represents  gravity,  Q 

the  mass  of  the  body,  and  D  its  semi-diameter.     Let  W  rep- 
resent weight  at  the  earth,  and  W  at  the  sun,  and  we  have 


W  :  W'  :  :       :  .  .  i  :  93.     Hence,  the  weight  of  a  body 

1  \\*2i 

at  the  sun  is  28  times  as  great  as  at  the  earth,  and  a  body 
would  fall  450  feet  in  the  first  second  of  its  descent. 

1O7.  Diurnal  rotation  of  the  sun.  —  By  observations  on  the 
solar  spots,  it  is  found  that  the  sun  rotates  on  its  axis  nearly  in 
the  same  direction  in  which  the  earth  revolves  about  the  sun. 
In  general,  a  spot  which  appears  on  the  edge  of  the  disk  passes 
across,  then  disappears,  and  afterward  reappears  in  the  same 
place  as  at  first  in  27£  days.  If  the  earth  were  at  rest,  this 
would  be  the  period  of  the  sun's 
rotation  on  its  axis.  But,  as  the 
earth  revolves  in  nearly  the  same 
direction  in  its  orbit,  the  appa- 
rent rotation  of  the  sun  is  longer 
than  its  real  rotation.  In  Fig. 
36,  suppose  the  earth  to  be  sta- 
tionary at  E,  and  that  a  spot  on 
the  sun  appears  on  the  disk  at 

A.  Then,  after  passing  through 

B,  D,  H,  it  will  appear  again  at 
A,  at  the  end  of  one  revolution. 

But,  if  the  earth  in  the  mean  time  moves  on  to  F,  then  the 


APPEARANCE   OF   THE   SOLAR  SPOTS.  65 

spot  must  pass  over  AB,  in  addition  to  one  revolution,  before 
it  will  be  seen  on  the  edge  of  the  disk.  As  EC  is  perpendicu- 
lar to  AD,  and  FC  to  BH,  the  corresponding  arcs  on  the  two 
circles  are  obviously  similar.  Therefore,  EGE  +  EF  :  EGE  : : 
ADA  +  AB  :  ADA.  Instead  of  the  arcs,  we  may  use  the 
times  of  describing  them ;  and  then  we  have  1  year  •+•  2TJ 
days  :  1  year  ::  27  J  days  :  25  days,  8  hours,  which  is  the  true 
period  of  the  sun's  rotation 

1O8.  Position  of  the  surfs*  equator. — If  the  solar  spots  al- 
ways described  their  paths  across  the  disk  in  apparent  straight 
lines,  it  would  be  inferred  that  the  sun's  equator  coincides 
with  the  plane  of  the  ecliptic.  But  these  lines  appear  straight 
only  twice  in  the  year,  near  the  middle  of  June  and  of  December. 
At  other  times,  they  appear  as  semi- ellipses,  having  the  greatest 
breadth  in  March  and  September.  The  earth,  therefore,  passes 
the  plane  of  the  sun's  equator  in  June  and  December.  The 
inclination  of  the  sun's  equator  to  the  plane  of  the  ecliptic  is 
found  to  be  about  Yi°. 

1O9. — Appearance  of  the  solar  spots. — On  examining  the 
sun's  disk  with  a  telescope,  there  is  usually  seen  a  greater  or  a 
less  number  of  dark  spots,  differing  from  each  other  in  form 
and  size,  and  each  spot  generally  consisting  of  two  distinct 
parts,  called  the  macula,  or  nucleus,  and  the  umbra.  The 
macula  is  black,  of  irregular  form,  and  commonly  surrounded 
by  the  umbra,  which  has  a  lighter  shade.  The  two  parts  of 
the  spot  do  not  often  shade  into  each  other,  but  are  each 
marked  by  a  sharp,  though  irregular  outline.  If  watched  from 
day  to  day,  they  are  seen  not  only  to  move  slowly  across  the 
disk,  as  already  stated,  but  they  change  their  form  and  general 
appearance.  A  large  spot  sometimes  divides  into  two  or  more 
smaller  ones ;  and  again  a  group  unites  into  a  single  large  spot. 
Sometimes  a  spot  diminishes  and  disappears,  first  the  macula, 
then  the  umbra.  The  reverse  also  happens — a  spot  is  seen  in 
the  midst  of  the  disk,  where  there  was  none  the  day  before. 
Though  only  a  few  are  commonly  in  sight  at  once,  yet  they  have 
been,  in  some  instances,  counted  by  tens  and  even  hundreds. 
Very  rarely  a  spot  is  so  large  as  to  be  seen  by  the  naked  eye. 

5 


66       RELATION  OF  SPOTS  TO  SURFACE  LEVEL. 

Figure  37  (lower  part)  shows  two  views  of  the  same  group, 
as  seen  July  9th  and  llth,  1 844. 

Fig.  37. 


July  9  July  11 


1 1 0.  The  spots  are  at  the  surface,  and  limited  to  a  northern 

and  a  southern  zone. — Each  spot  appears  on  the  disk  during  one- 
half  the  time  of  its  entire  revolution.  It  must,  therefore,  be  at 
the  surface,  and  not  at  any  distance  from  it.  For,  if  it  revolved 
at  any  distance  from  the  surface,  as  in  the  orbit  abc  (Fig.  35), 
then  it  would  be  seen  on  the  disk  only  from  a  to  5,  which  is 
less  than  half  its  orbit. 

But  the  spots  do  not  pass  across  all  portions  of  the  disk; 
their  paths  are  limited  to  a  zone  which  extends  not  more  than 
35°  on  each  side  of  the  equator ;  and  with  very  few  exceptions, 
they  lie  in  the  outer,  rather  than  the  central  parts  of  this  zone. 
Spots  are  very  rarely  seen  within  the  zone  lying  between  10° 
of  north  and  south  latitude;  and  still  more  rarely  in  the  polar 
zones  above  latitudes  35°  north  and  35°  south.  The  macular 
zones,  as  they  are  sometimes  called,  are  represented  in  Figure 
37,  limited  by  the  dotted  lines,  EQ  being  the  equator. 

111.  Relation  of  the  spots  to  the  surface  level. — If  the  spots 
were  flat  surfaces  on  the  same  level  with  the  general  surface 
of  the  sun,  then  all  their  parts  would  be  foreshortened  alike, 


THE   RECEIVED   THEORY.  67 

when  near  the  edges  of  the  disk.  If  they  were  elevated  ob- 
jects, as  mountains,  rising  above  the  solar  atmosphere,  then  the 
umbra  nearest  the  edge  of  the  disk  would  be  hidden  by  the 
darker  part,  and  on  the  edge  the  spot  would  appear  as  a  pro- 
tuberance. 

But  it  is  proved,  by  multiplied  observations,  that  the  spots 
must  be  depressions  below  the  general  surface,  and  the  macula 
a  deeper  depression  than  the  umbra.  For,  as  a  spot  approaches 
the  edge,  while  it  is  foreshortened  by  perspective,  the  umbra 
furthest  from  the  edge  disappears  first,  and  then  the  macula 
itself,  while  that  part  of  the  umbra  nearest  the  edge  is  still  in 
sight.  As  a  spot  conies  from  the  edge  toward  the  central  part 
of  the  disk,  the  order  of  appearances  is  reversed.  These 
changes  are  indicated  in  the  upper  zone  of  Figure  37. 

112.  The  general  surface. — The  luminous  part  of  the  sun's 
surface  is  not  uniform,  nor  at  rest.     Every  portion  of  it  is  mi- 
nutely mottled  by  spots  and  streaks  of  unequal  illumination. 
These  are  called  faculce.     And  continued  observation  showrs 
that   these   faint   inequalities   are   also   undergoing    incessant 
changes.     The  faculae  are  most  strongly  marked,  and  indicate 
the  greatest  agitation  of  surface,  where  a  spot  is  about  to  ap- 
pear, or  where  one  has  recently  disappeared. 

113.  The  received  theory. — !Nb  theory  so  well  explains  the 
telescopic  appearances  of  the  sun,  as  that  which  in  substance 
was  proposed  by  Sir  William  Herschel,  in  1801.     Whatever 
may  be  the  condition  of  the  central  mass,  the  external  surface, 
called  the  photosphere,  is  a  gas  in  a  state  of  incandescent  flame, 
while   below   it,   within   the    solar    atmosphere,   is   a   cloudy 
stratum,  less   luminous   than  the    outer  surface.     Whenever, 
from  any  cause,  a  rent  is  made  in  the  photosphere,  the  less 
luminous  stratum  below  is  seen  through  it,  as  the  umbra  of  a 
spot;    and  a  smaller  rent  in  the  lower  stratum  reveals  the 
denser  and  darker  part  of  the  sun,  as  the  macula  of  the  same 
spot.     The  strata  in  which  the  rents  occur  are  in  a  gaseous 
condition ;  for  the  constant  motions  going  on  in  the  outlines 
of  the  spots,  forbid  the  supposition  that  they  consist  of  solid 
matter ;  and  the  extreme  rapidity  of  these  motions,  often  more 


68  THE   ZODIACAL  LIGHT. 

than  1,000  miles  per  day,  is  inconsistent  with  the  idea  that 
they  are  liquid. 

114.  The  l>ody  of  the  sun  not  necessarily  dark. — The  very 
dark  appearance  of  the  macula  may  be  due  to  its  strong  con- 
trast  with  the   intense   illumination   of  the   general    surface. 
For  it  is  found  by  experiment  that  the  brightest  artificial  light 
which  has  been  produced,  if  placed  between  the  eye  and  the 
sun,  appears  as  a  dark  spot  compared  with  the  solar  surface. 

115.  Cause  of  the  spots. — Sir  John  Herschel  has  suggested 
that  there  are  reasons  for  considering  the  equatorial  regions  of 
the  sun  to  be  more  heated  than  the  other  portions,  so  that  there 
are  currents  in  the  solar  atmosphere  analogous  to  the   trade- 
winds' on  the  earth.     Resulting  from  these  currents,  he  sup- 
poses that  occasional  local  winds  are  produced,  rotating  on  a 
vertical  axis,  and  rending  the  atmosphere  and  clouds  by  their 
centrifugal  force.     The  ruptures  thus  occasioned  are  the  spots 
on  the  sun. 

This  supposition  derives  considerable  plausibility  from  the 
considerations,  that  the  spots  are  limited  to  narrow  zones  a  little 
distance  from  the  equator;  that  they  sometimes  differ  from 
each  other  in  their  motions  across  the  disk;  and  that,  in  a  few 
instances,  they  have  shown  signs  of  rotation  about  their  own 
centers  i 

116.  Periodicity  of  the  spots. — The  number  and  size  of 
spots  vary  exceedingly  in  different  years.    Sometimes  for  days 
and  weeks  none  are  to  be  seen  ;  and  again,  for  many  months, 
the  disk  is  never  free  from  them.     It  is  noticed,  of  late  years, 
that  their  frequency  alternately  increases  and  decreases  during 
a  period  of  10  or  11  years.     The  years  in  which  the  greatest 
number  has  been  seen  of  late,  were  1828,  1838,  1849,  1860. 
And  those  in  which  there  were  fewest,  were  1833,  1843,  1854. 

117.  The  zodiacal  light. — This  name  is  given  to  a  faint,  ill- 
defined  light,  extending  along  the  zodiac,  either  in  the  west, 
after  sunset,  or  in  the  east,  before  sunrise.     It  so  much  resem- 
bles the  twilight,  that  it  is  not  ordinarily  noticed,  because  it 


69 

appears  as  a  mere  upward  extension  of  it.  It  is  projected  on 
the  sky  as  a  triangle,  inclined  to  the  horizon  at  the  same  angle 
as  the  ecliptic  (Fig.  38).  In  the  Fi  3g 

evening  it  is  best  seen  at  the 
season  when  the  ecliptic  is  most 
nearly  perpendicular  to  the  ho- 
rizon, after  twilight  has  ceased. 
It  is  therefore  most  conspicu- 
ous at  evening  in  the  month 
of  February.  When  the  air 
is  clear,  and  there  is  no  moon, 
it  is  visible  till  after  9  o'clock. 
For  a  like  reason,  the  best 
time  for  seeing  it  before  morn- 
ing twilight  is  the  month  of 
October.  The  apparent  extent 
of  it,  both  in  breadth  and 
height,  is  much  increased  by 
indirect  vision. 

\  18.  Its  nature. — There  has  been  much  speculation  rela- 
tive to  the  nature  of  the  zodiacal  light.  But  astronomers  gen- 
erally regard  it  as  a  nebulosity  attending  the  sun,  and  extend- 
ing beyond  the  orbits  of  Mercury  and  Yenus,  and  even  beyond 
the  orbit  of  the  earth. 


CHAPTER  VIII. 

KEPLER'S  LAWS. — THE  LAW  OF  GRAVITATION. 

119.  Statement  of  Kepler*  slaws. — From  a  long  and  labo- 
rious examination  of  the  recorded  observations  of  Tycho  Brahe, 
Kepler  deduced  three  laws  relating  to  the  movements  of  the 
planets  about  the  sun.  They  are  hence  called  Kepler's  laws, 
and  may  be  stated  as  follows. 

1.  The  areas  described'  about  the  sun  l)y  the  radius  vector  of 
an  orbit*  vary  as  the  times  of  describing  than. 


70 


LAW    OF    AREAS. 


2.  The  orbit  of  every  planet  is  an  ellipse,  having  the  sun  in 
one  focus. 

3.  The  squares  of  the  periodic  times  of  the  several  planets 
vary  as  the  cubes  of  their  mean  distances. 

To  render  the  language  of  the  third  law  strictly  correct,  the 
cube  of  the  distance  should  be  divided  by  the  sum  of  the  masses 
of  the  sun  and  planet.  But  the  mass  of  even  the  largest  planet 
is  so  small,  compared  with  the  sun,  that  the  omission  intro- 
duces an  error  which  is  scarcely  appreciable. 

Kepler  established  these  three  laws  as  facts  in  the  solar 
system  ;  but  Newton  afterward  demonstrated,  by  mathematical 
reasoning,  that  they  are  necessarily  involved  in  the  laws  of  in- 
ertia and  gravitation. 

1 2O.  Areas  described  by  the  radius  vector. — "Whatever  path 
a  body  describes  under  the  influence  of  a  projectile  and  a  cen- 
tripetal force,  the  areas  described  about  the  center  of  force  vary 
as  the  times  of  describing  them. 


Let  S  (Fig.  39)  be  the  center  of  attraction,  and  suppose  the 
projectile  force  in  the  line  YE.  to  be  such  as  to  cause  the  body 
to  pass  over  the  equal  spaces  PQ,  QK,  etc.,  each  in  a  certain 


LAW   OF   VELOCITY   IN   AN   OEBIT.  71 

unit  of  time.  "When  the  body  reaches  Q,  let  the  action 
toward  S  be  sufficient  to  move  it  over  QY  in  the  same  time 
in  which  by  the  original  impulse  it  would  describe  QE.  Then 
it  will  in  the  same  time  describe  the  diagonal  QC  of  the  par- 
allelogram. Join  ES  and  CS.  The  triangles  QSC  and 
QSE  are  equal  ;  but  QSE  =  QSP  ;  .-.  QSC  =  QSP,—  that  is, 
the  areas  described  in  the  first  and  second  units  of  time  are 
equal.  In  like  manner,  by  supposing  a  second  action  toward 
S  to  occur  at  C,  a  third  at  D,  etc.,  it  is  proved  that  QCS,  CDS, 
DES,  etc.,  which  are  described  in  equal  times,  are  equal.  This 
is  true,  howrever  small  the  unit  of  time  between  the  successive 
actions  toward  S,  and  is  therefore  true  when  the  central  force 
acts  incessantly  and  causes  curvilinear  motion.  As  all  the 
areas  are  equal,  which  are  described  in  the  'several  units  of 
time,  therefore  the  areas  vary  as  the  times. 

As  the  diagonal  of  each  parallelogram  is  in  the  same  plane 
with  its  two  sides,  it  is  obvious  that  the  whole  orbit  lies  in  one 
and  the  same  plane. 

Conversely,  if  areas  described  about  a  point  'Gary  as  the 
times,  the  deflecting  force  acts  toward  that  point.  For 
PSQ  =  QSE,  as  before  (Fig.  39)  ;  and  by  supposition,  PSQ  = 
QSC  ;  /.  QSC  =  QSE  ;  hence  CE  is  parallel  to  QS,  and  QC  is 
the  diagonal  of  a  parallelogram,  whose  side  QY,  in  which  the 
deflecting  force  acts,  is  directed  toward  S. 

Since  it  is  an  established  fact,  agreeably  to  Kepler's  first  law, 
that  the  radius  vector  of  each  planetary  orbit  describes  areas 
about  the  sun,  which  vary  as  the  times;  therefore,  the  cen- 
tripetal force,  acting  on  the  planets,  is  directed  toward  the  sun. 

121.  The  law  of  velocity  in  an  orbit.  —  The  velocity  at  any 
point  varies  inversely  as  the  perpendicular  from  the  center  of 
force  to  the  tangent  at  that  point. 

Let  SY  (Fig.  39)  be  perpendicular  to  PQ  \   then  the  area 


SPQ  =iPQ  x  SY,  which  varies  as  PQ  x  SY  ;  .-.  PQ  oo  -. 

b  I 

But  PQ  oo  Y,  the  velocity  at  P  ;  and  the  area  SPQ  is  constant; 
,  •.  Y  co  ~?  or  the  velocity  varies  inversely  as  the  perpendicu- 


72 


LAW   OF   GRAVITATION. 


lar  from  S,  upon  the  line  in  which  the  body  is  moving;  in 
other  words,  upon  the  tangent  of  its  path,  if  it  describes  a 
curve. 

122.  Law  of  gravitation  in  an  orbit,  as  related  to  dis- 
tance.— If  a  body  describes  an  elliptical  orbit,  by  a  centripetal 
force  which  acts  toward  the  focus,  tliat  force  varies  inversely  as 
the  square  of  the  distance. 


Fig.  40. 


Let  the  body  be  at  M  (Fig.  40),  and  MF  the  radius  vector  at 
that  point.  Let  MO  be  the  radius  of  curvature  at  M,  and 
therefore  perpendicular  to  the  tangent ;  and  suppose  MN"  to  be 
an  infinitely  small  arc  described  in  a  given  small  portion  of 
time.  Draw  FP  perpendicular  to  the  tangent  MP,  OT£  to 
FM,  and  NH  to  MO ;  then  PFM,  MHI,  KNI  are  similar  tri- 
angles. MN,  considered  as  a  straight  line,  is  described  by  the 
joint  action  of  the  centripetal  force  MI,  and  the  projectile 
force,  which  is  equal  and  parallel  to  IN.  The  motion  in  MI 
may  be  regarded  as  uniformly  accelerated,  because  in  the  in- 
finitely small  time  of  describing  it,  the  centripetal  force  may 
be  considered  constant.  Hence,  2MI  may  be  taken  as  the 
measure  of  the  centripetal  force  f  (Nat.  Phil.,  Art.  28). 

Therefore,  f  oo  MI.    It  is  to  be  proved  that  MI  oo 


LAW   OF   GKAVITATION.  73 

123.  By  similar  triangles,  MI  :  MH  :  :  NI  :  NK; 


. 

Now,  the   chord   MN   is  a  mean  proportional  between   the 

MN2 
versed   sine  MH  and   the   diameter  2MO  ;   or  Mil  =  OT 


but,  as  the  arc  is  infinitely  small,  NH  =  MN ;  .*.  MH  =       ^-. 

Again,  the  versed  sine  MH,  and  therefore  HI,  is  infinitely 
small  compared  with  NH,  and  NI  may  be  substituted  for 

NI2 

•••^=So- 

124.  Now  it  is  shown  in  conic  sections,  that 

~2  \FP/  ' 

FM      NI 

therefore,  since  by  similar  triangles  -™  = 


'  2  VNK 
Substituting  this  for  MO  in  the  equation  for  MH  above,  we  have 


p  .  NI 
Hence,  in  the  equation  for  MI  we  have 

NK3        NI        1  .,      - 
MI  =  —  ===  x  ==-  =  -  NK  . 
p  .  NI      NK      p 

Now,  the  sector  FMN  is  measured  by  JFM  .  NK  ;  .  •  .  NK  = 

*  Jackson's  Conic  Sections.     The  same  may  be  derived  from  Coffin's  Conic 
Sections,  Pr.  V.,  Curvature,    R2  or  MO  =  '  -  *—  —  '  ,  a  and  b  being  the  semi- 

axes  ;  .-.  MO  =   -  x  (FM  .  MV)2".    Multiply  by  (&2)  *,  and  divide  by  its  equal 


=  ,  since  FMP  and  VML 


T,   .  6*        p          T.Trk      p    /FM2\1        p    /FM\3 
are  smJar.     But  _  =       .  ,.  MO  =     -  •  =  . 


74:  LAW   OF   GKAVITATION. 

2FMN  4FMN2  4FMN2 

~^;  •'•  MI    =  2'     But  as  the 


areas  described  by  the  radius  vector  vary  as  the  times,  FMN  is 
constant.     Therefore, 

MI  (=/)«, 


that  is,  the  centripetal  force  in  the  orbit  varies  inversely  as  the 
square  of  the  distance. 

125.  Applicable  to  every  conic  section.  —  It  is  thus  proved 
that,  in  any  elliptical  orbit  described  about  the  focus  as  the 
center  of  attraction,  the  intensity  of  that  attraction  varies  in- 
versely as  the  square  of  the  radius  vector.     As  there  is  nothing 
in  the  foregoing  demonstration  to  limit  the  conclusion  to  the 
orbits  which  are  nearly  circular,  like  those  of  the  planets,  we 
are  at  liberty  to  apply  it  to  orbits  of  extreme  eccentricity,  as 
those  of  the  comets.     And  it  is  proved  by  Newton,  in  his  Prin  • 
cipia,  that  the  same  law  of  force  is  necessary,  in  order  that  a 
body  may  describe  any  one  of  the  conic  sections  about  its  focus 
as  the  center  of  attraction. 

126.  Law  of  gravitation  as  to  distance,  in  different   or- 
bits. —  And  not  only  does  this  law  prevail  in  all  parts  of  any 
one  orbit,  but  it  is  true  also  that  all  the  different  bodies  of  a 
system,  describing  orbits  about  the  same  center  of  force,  are 
urged  toward  that  center  by  attractions  which  vary,  from  one 
orbit  to  another,  inversely  as  the  square  of  the  distance. 

Let  a  be  the  semi-major,  and  b  the  semi-minor  axis  of  any 
elliptic  orbit.  Then  a  is  the  mean  distance  of  all  points  of  the 
orbit  from  the  focus.  By  a  rule  of  mensuration,  the  area  of 
the  ellipse  =  nab.  If  s  =  the  area  described  by  the  radius 
vector  in  a  unit  of  time,  as  one  second,  and  t  =  the  number  of 
seconds  in  the  whole  period  of  revolution,  then  the  ellipse  also 

=  ts.     Therefore,  nab  =  ts  ;  and  t  =  -  -  ;  and  £2  =  —  —  .    By 

s    '  s 

272  Z  2 

Kepler's  third  law  (Art.  119),  £2oo  a3  ;   .'.  —5-  oo  a3  ;  /.  —  oo  s2. 

S  Ob 

ID 

But,  because  the  semi-parameter  4-  is  a  third  proportional  to 


LAW    OF   GRAVITATION. 


75 


the  semi-axes  a  and  £,  —  =  ^  ;   .  * .  -|-  oo  /.     Hence,  substituting 

a  a  -j 

^  for  s2,— that  is,  FMN2? — in  the  equation  for  MI  (Art.  124),  we 


find   MI  =  ^^  =  _^p  =  j±p  ;    .-.  /«  j^p.      Or,    the 

force  varies  inversely  as  the  square  of  the  distance,  in  different 
orbits,  as  well  as  in  different  parts  of  the  same  orbit. 

The  satellites  which  revolve  about  the  planets  are  found  to 
conform  to  Kepler's  laws,  and  therefore  the  force  which  urges 
them  toward  their  respective  primaries  varies  in  each  case  in- 
versely as  the  square  of  the  distance. 

127.  Law  of  gravitation  within  small  distances. — But  the 
inquiry  still  remains,  does  the  law  of  gravity,  as  demonstrated 
in  the  foregoing  articles,  hold  good  at  the  smallest  distances 
also  ?  For  example,  do  the  tendencies  of  bodies  resting  on  the 
earth,  and  of  those  elevated  in  the  air,  and  of  the  moon  toward 
the  earth's  center,  come  under  the  same  general  law  ?  This  is 
the  very  question  which  presented  itself  to  the  mind  of  New- 
ton, after  he  had  discovered  that  the  force  which  deflects  the 
planets  from  their  lines  of  motion  toward  the  sun,  varies  in- 
versely as  the  square  of  their  distance  from  it.  As  he  noticed 
the  fall  of  an  apple,  the  inquiry  arose,  may  not  this  fall  be  of 
the  same  nature  as  the  lending  of  the  moon's  path  toward  the 
earth,  and  may  not  the  force  in  the  two  cases  be  as  the  squares 
of  the  distances  inversely  ?  Fi  41 

The  distance  through  which  the 
moon  actually  descends  in  one  second 
may  be  represented  by  A.a  (Fig.  41), 
A I  being  the  arc  described  in  the  same 
time.  For,  as  the  moon  was  going 
toward  B,  it  would  not  have  deviated 
fr-om  the  line  AB,  if  some  force  had 
not  turned  it  aside.  This  influence 
must  be  directed  toward  the  earth,  E, 
because  it  is  about  E  that  the  radius 
is  known  to  describe  areas  propor- 
tional to  the  times  (Art.  120).  There- 


76  LAW    OF    GRAVITATION. 

fore,  B5,  or  the  versed  sine  Aa  (which  may  be  considered  equal 
to  it),  is  the  distance  fallen  through  in  one  second.  Now,  the 
circumference  of  the  moon's  orbit,  divided  by  the  number  of 
seconds  occupied  in  describing  it,  gives  the  arc  Ah.  This  arc 
and  its  chord  may  be  considered  the  same,  and  by  geometry 
we  have  2  AE  :  A!J  ::  Ah  :  Aa  =  0.0535  of  an  inch. 

At  the  surface  of  the  earth,  a  body  falls  1 6^3  feet  in  the 
first  second.  On  the  supposition  that  gravity  varies  inversely 
as  the  square  of  the  distance,  we  find  the  fall  in  one  second  at 
the  moon,  by  the  proportion,  the  square  of  the  moon's  dis- 
tance :  square  of  the  earth's  radius  : :  1 6^3  feet  :  0.0536  of  an 
inch,  agreeing  very  accurately  with  the  distance  which  the 
inoon  actually  falls  from  a  tangent  in  one  second.  Therefore, 
a  body  falling  at  the  surface  of  a  planet,  and  a  satellite  revolv- 
ing about  it,  are  both  subject  to  the  same  law  of  centripetal 
force. 

128.  The  law  prevails  throughout  the  solar  system. — As 
will  appear  hereafter,  there  are  numerous  disturbances  pro- 
duced upon  the  motion  of  each  body  in  the  system  by  the 
attraction  of  every  other.     Every  one  of  these  disturbing  influ- 
ences is  measured,  by  applying  the  law  of  distance  already  men- 
tioned.    If  a  planet  or  comet  moves  toward  a  planet  for  a 
certain  length  of  time,  it  is  accelerated  ;  and  its  acceleration  is 
greater,  as  the  square  of  the  distance  is  less ;  and  it  is  retarded, 
according  to  the  same  law,  when  departing  from  it. 

129.  The  law  of  gravitation,  as  related  to  the  quantity  of 
matter. — The  force  of  gravity  varies  directly  as  the  quantity  of 
matter.     In  Mechanics,  we  infer  the  existence  of  this  law  from 
the  fact  that  all  bodies,  light  and  heavy,  and  of  every  kind  of 
material,  fall  with  equal  velocity  toward  the  earth.     So,  in  the 
solar  system,  a  planet  and  all  its  satellites,  when  at  equal  dis- 
tances from  the  sun,  are  urged  towrard  it  by  forces  proportional  to 
their  masses,  or  they  could  not  maintain  their  mutual  relations 
as  they  do.     And  it  is  found  that  every  disturbing  influence  in 
the  pystem  is  accounted  for  only  by  applying  both  parts  of  the 
law  of  gravity — that  it  varies  directly  as  the  quantity  of  matter, 
and  Inversely  as  the  square  of  the  distance. 


PATHS   OF   PROJECTILES. 


77 


130.  Paths  of  projectiles  considered  as  orlits. — When  a 
stone  is  thrown,  or  a  ball  is  fired,  its  path  (undisturbed  by  the 
atmosphere)  is  part  of  an  elliptic  orbit,  one  of  whose  foci  is  at 
the  center  of  the  earth.     In  Mechanics,  the  path  of  a  projectile 
is  proved  to  be  a  parabola  (Nat.  Phil.,  Art.  49) ;  but,  in  that 
demonstration,  the  vertical  lines  were  assumed  to  be  parallel 
to  each  other,  and  the  force  of  gravity  a  constant  force,  neither 
of  which  is  strictly  true.     Knowing  the  distance  and  period  of 
the  moon,  the  time  in  which  a  projectile  would  complete  its 
revolution  is  found  by  Kepler's  third  law.     Any  force,  which 
man  could  apply,  would  carry  the  lower  extremity  of  the  orbit 
so  little  beyond  the  center  of  the  earth,  that  the  mean  distance 
might  be  called  one-half  the  radius  of  the  earth.     Therefore, 
calling  the  moon's  distance  60  radii,  and  its  period  27J  days, 
we  have  (60)3  :  (|)3  ::  (27|)2  :  tf,  from  which  x  is  found  to  be 
about  31  minutes.     Every  projectile,  then,  if  it  were  free  to 
complete  its  orbit  unobstructed,  and  according  to  the  law  of 
gravity  which  prevails  outside  of  the  earth,  would  make  an  en- 
tire revolution,  and  return  to  its  place,  in  about  half  an  hour. 

131.  Effect  of  increased  velocity  of  projection. — Suppose 
that  P  (Fig.  42)  is  a  point  near  the  earth,  ADE,  and  that  the 
velocity  of  projection,  in  the  direction  PB,  is  so  greatly  in- 
creased that  the  projectile  strikes  the  earth  at  D.     By  a  still 
greater  increase  of  velocity 

it  might  meet  the  earth  at  E. 
In  these  cases  the  earth's 
center  would  be  in  the  most 
remote  focus  of  the  orbit. 
But  if  we  suppose  the  velo- 
city so  much  increased  that 
the  centrifugal  force  just 
equals  the  force  of  gravity, 
then  the  body  would  de- 
scribe the  circular  orbit  PFGr 
(Art.  90).  As  the  mean  dis- 
tance now  equals  the  radius 
of  the  earth,  the  time  of  revolution  is  found,  by  Kepler's  third 
law.  to  be  Ih.  24m.  39s.  Any  increase  of  the  velocity  of  pro- 


78  MOTIONS  OF  SUN  AND   PLANET. 

jection  beyond  this  will  again  produce  an  ellipse,  as  PK, 
whose  nearer  focus  is  at  the  earth's  center.  And  we  can 
imagine  the  velocity  increased  till  the  ellipse  becomes  one  of 
extreme  eccentricity,  and  then  changes  into  the  branch  of  a 
parabola,  and  then  of  a  hyperbola,  in  which  last  cases  the  body 
will  never  commence  a  return  toward  the  earth. 

132.  Orbit  motion  and  diurnal  rotation  by  one  impulse. — 
If  we  suppose  the  projectile  motion  of  the  earth,  or  any  other 
planet,  to  have  been  produced  by  a  single  impulse,  that  im- 
pulse may  also  have  caused  the  diurnal  rotation  of  the  body. 
If  the  impulse  had  been  directed  in  a  line  passing  through  the 
center  of  gravity  of  the  planet,  then  it  would  have  caused  a 
progressive  motion  without  rotation  on  an  axis.     But,  if  the 
line  of  impulse  did  not  pass  through  the  center  of  gravity, 
there  would  be  rotation  as  well  as  progression.     It  has  been 
calculated  that  the  two  existing  rotations  of  the  earth  might 
have  been  produced  by  one  impulse,  applied  in  a  line  which 
passes  24  miles  from  the   earth's   center,   on  the  side  most 
remote  from  the  sun. 

Had  it  been  directed  through  a  point  lying  on  the  side 
nearest  the  sun,  the  diurnal  motion  would  obviously  have  been 
retrograde. 

133.  Motions  of  sun  and  planet,  resulting  from  an  impulse 
given  to  the  planet. — Suppose  that  the  sun  at  S  (Fig.  43),  and 
the  earth  at  E,  mutually  attract  each  other,  and  that  an  im- 
pulse is  given  to  E  in  a  line  perpendicular  to  ES.     S  can  not 
remain  stationary  and  E  revolve  about  it ;  for  it  is  proved  (Nat. 
Phil.,  Art.  89)  that  their  center  of  gravity  will  move  precisely 
as  the  sum  of  the  bodies  would  move  if  united  at  the  center, 
and  the.  same  impulse  were  applied  to  them.     Suppose,  for  the 
sake  of  simplicity,  that  the  weights   of  the  bodies   and  the 
strength  of  the  impulse  are  so  related  that  the  center,  C,  will 
pass  over  each  unit  of  space,  C#,  ab,  be,  etc.,  while  E  advances 
45°  in  a  circle   about   the  moving  center.     Then,  when  the 
center  is  at  «,  E  is  at  1,  45°  from  a  perpendicular  at  a.     But  S 
must  be  on  the  opposite  side  of  #,  and  as  far  from  it  as  from  C 
before.     Therefore,  by  the  impulse  given  to  E.  and  the  mutual 


A  PLANET   AT   APHELION   OR  PERIHELION. 


79 


attraction  between  E  and  S,  the  latter  lias  been  drawn  along 
from  S  to  r.  Again,  when  the  center  is  at  ft,  E  is  at  2,  and  S 
at  2'.  While  E  was  on  the  upper  side  of  C/>,  S  was  drawn 
toward  that  line,  and  now  crosses  it.  and  by  its  inertia  con- 
tinues upward,  although  E  is  now  below  the  line.  In  this 
manner  the  bodies  revolve  about  the  moving  center,  describing 
circles  relatively  to  that,  but  curves  of  a  totally  different  char- 
acter in  space.  These  curves  are  always  some  variety  or  other 
of  the  class  of  curves  called  epicycloids.  In  the  case  repre- 
sented in  the  figure,  the  planet  describes  an  epicycloid  which 
forms  a  series  of  loops,  intersecting  its  own  path  at  every  revo- 
lution, while  the  path  of  the  heavier  body  is  of  a  waving  form. 
The  body  E  retrogrades  on  the  lower  part  of  the  loop  from  3 
to  5,  while  S  advances  continually,  but  with  unequal  velocities, 
each  body  being  alternately  drawn  forward  and  held  back  by 

the  other. 

Fig.  43. 


The  only  way  in  which  two  separate  bodies  could  be  made  to 
rotate  about  a  fixed  center  of  gravity,  would  be  to  give  an 
equal  impulse  to  each  body,  and  in  opposite  directions.  Two 
such  forces  would  constitute  a  couple  (Nat.  Phil.,  Art.  57), 
whose  effect  is  to  produce  rotation  merely. 

134.  Why  a  planet  at  aphelion  begins  to  return ,  or  at  peri- 
helion begins  to  depart. — It  might  be  thought  that  a  planet  at 


80 


PRECESSION   OF   EQUINOXES. 


its  aphelion,  C  (Fig.  44),  being  less  attracted  toward  the  sun 
than  at  any  other  point,  would  continue  to  withdraw,  instead 
of  commencing  to  return  ;  and  that  when  at  its  perihelion,  G, 
being  more  attracted  than  else- 
where, it  would  continue  to  ap- 
proach till  it  falls  to  the  sun.  The 
reason  why  a  planet  begins  to  re- 
turn after  reaching  the  aphelion  is 
to  be  found  in  its  diminished  ve- 
locity. As  the  planet  recedes 
through  H,  K,  and  A,  the  centrip- 
etal force  toward  S  draws  it  back, 
and  causes  continual  retardation, 
till  at  C  the  velocity  is  so  much 
diminished  that  the  attraction  of  S, 
though  less  than  elsewhere,  is  still 
sufficient  to  curve  the  path  so  that  it  falls  within  a  circle  about 
the  centre  S,  and  the  planet  begins  to  approach  the  sun. 

Again,  as  the  planet  passes  through  D,  E,  and  F,  the  at- 
traction toward  S  partly  conspires  with  its  inertia,  and  it  is 
continually  accelerated,  till,  at  G,  its  velocity  has  become  so 
great  that  its  path  strikes  outside  of  a  circle  about  the  center, 
S,  and  it  begins  again  to  depart  as  before. 


CHAPTEE    IX. 

PEECESSION   OF   EQUINOXES. — NUTATION. — ABERRATION  OF 
LIGHT. — APSIDES   OF   THE   EARTH'S   ORBIT. 

135.  Precession  of  equinoxes  d>- scribed. — The  points  in 
which  the  equator  intersects  the  ecliptic  on  the  celestial  sphere 
are  not  stationary,  but  have  a  slow  retrograde  movement — that 
is,  they  revolve  from  east  to  west.  The  sun,  therefore,  in  its 
annual  progress  eastward,  crosses  the  equator  each  year  a  little 
further  west  than  it  did  the  year  previous.  This  motion  is 


CAUSE    OF    PEECESSION.  81 

called  the  precession  of  the  equinoxes,  either  because  the  time 
of  the  equinoxes  precedes  the  time  in  which  the  sun  would  have 
passed  them  if  they  had  remained  at  rest,  or  because,  in 
the  daily  transit  of  the  meridian,  the  equinoxes  precede  those 
stars  which  crossed  at  the  same  time  with  them  the  preceding 
year. 

The  equinoctial  points  retrograde  about  50£"  each  year.  At 
this  rate,  it  will  require  25,800  years  to  make  a  complete  circuit 
of  the  heavens. 

136.  Signs  of  the  ecliptic  displaced  from  the  signs  of  the 
zodiac. — The  want  of  coincidence  between  the  signs  of  the 
ecliptic  and  the  signs  of  the  zodiac  was  noticed  (Art.  61).    They 
coincided  at  the  time  the  division  was  made,  about  2,000  years 
ago ;  and  the  precession  daring  this  period  has  moved  the  equi- 
noxes backward  2,000  x  50i"  =  28°,  nearly.     Hence,  Aries  of 
the  zodiac  almost  coincides  with  Taurus  of  the  ecliptic,  Taurus 
of  the  zodiac  with  Gemini  of  the  ecliptic,  etc. 

137.  Motion  of  the  north   and  south  poles. — Considering 
the  plane  of  the  ecliptic  as  fixed,  its  poles  of  course  occupy 
fixed  positions  among  the  stars.     But  this  is  not  true  of  the 
poles  of  the  equator.     Their  distance  from  the  poles  of  the 
ecliptic  is  equal  to  the  obliquity  of  the  two  circles — that  is, 
23°  27'.     As  this  angle  remains  nearly  constant,  and  the  points 
of  intersection  move  around  westward,  the  poles  of  the  equator 
must  likewise  move  round  those  of  the  ecliptic  in  the  same 
direction,  and  occupy  the  same  period,  25,800  years  in  com- 
pleting their  revolution.     The  north  pole  of  the  equator  is  now 
near  the  star  in  Ursa  Minor,  known  as  the  pole-star.     Accord- 
ing to  the  earliest  catalogues,  the  pole  was  12°  distant  from  the 
pole-star.     It  is  now  somewhat  more  than  1°  distant,  and  will, 
at  the  nearest,  pass  within  £°  of  it.     In  about  13,000  years  the 
pole  will  be  on  the  opposite  side  of  the  pole  of  the  ecliptic,  near 
the  bright  star  a  Lyrse,  which  will  then  be  the  pole-star. 

138.  Cause  of  precession. — The  precession  of  the  equinoxes 
is  a  disturbance  produced  by  the  sun's  and  moon's  attraction 
upon  the  equatorial  ring  of  the  earth,  as  it  rotates  on  its  axis. 

6 


82  CAUSE    OF    PKECESSION. 

The  sun  being  in  the  ecliptic,  while  the  equatorial  ring  is  in- 
clined 23°  27'  to  it,  the  sun's  attraction  is  oblique  to  the  plane 
of  the  ring ;  and  one  component  of  this  force  is  perpendicular 
to  the  ecliptic.  This  component  tends  to  turn  the  nearer  half 
of  the  ring  on  the  line  of  equinoxes  as  a  hinge  toward  the 
ecliptic.  The  remote  half  is  pressed  from  the  ecliptic,  but  not 
BO  much  as  the  nearer  half  is  toward  it ;  so  that  the  disturb- 
ance, on  the  whole,  is  toward  the  ecliptic.  And  this  tendency, 
compounded  with  the  inertia  of  the  ring  in  its  diurnal  rotation, 
moves  the  equinoxes  backward. 

Fig.  45. 


Let  EC  (Fig.  45)  represent  the  plane  of  the  ecliptic,  and 
QK  the  equatorial  ring  of  matter.  A  particle,  A,  of  the  ring, 
by  its  inertia  of  rotation,  tends  to  move  toward  T  in  the  plane 
QK.  Let  AB  represent  this  force,  and  AF  the  pressure  toward 
EC,  produced  by  the  sun ;  then  the  resultant  will  be  the  diag- 
onal AD,  shifting  the  equinox  back  to  T'.  All  the  particles 
are  subjected  to  this  influence,  except  at  the  moment  (each  day) 
of  crossing  T  and  — ,  so  long  as  the  sun  itself  is  not  in  the  line 
Y=£=  produced,  which  occurs  in  March  and  September.  The 
effect  is  then  interrupted  for  a  time. 

As  the  moon  is  always  near  the  ecliptic— sometimes  on 
one  side  of  it,  and  sometimes  on  the  other — its  action  on  the 
whole  conspires  with  that  of  the  sun.  And  as  it  is  compar- 
atively near,  though  it  is  so  small  a  body,  its  effect  is  more 
than  twice  as  great  as  that  of  the  sun.  The  planets  produce  a 


THE    TROPICAL    AND    SIDEREAL    YEAH.  83 

very  minute  effect  on  the  ring,  tending  to  dimmish  the  amount 
of  precession.  The  joint  effect  of  all  the  bodies  mentioned  is, 
as  stated  above,  50J". 

1  39.  Law  of  composition  of  rotations. — The  case  of  pre- 
cession of  equinoxes  is  classed  under  the  general  law  for  the 
composition  of  two  rotations,  which  is  analogous  to  that  for 
the  composition  of  two  rectilinear  motions  (Nat.  Phil.,  Art. 
42).  It  may  be  stated  thus :  if  two  forces  are  applied  to  a 
body,  which,  separately,  would  cause  rotation  on  two  different 
axes,  their  joint  action  will  produce  rotation  on  a  third  axis 
lying  in  the  plane  of  the  other  two,  and  making  angles  with 
them,  whose  sines  are  inversely  as  the  forces.  In  precession, 
the  earth  rotates  on  the  diurnal  axis  by  one  force,  and  the  sun 
and  moon  tend  to  rotate  it  on  the  line  of  the  equinoxes.  As 
the  latter  force  is  minute  compared  with  the  other,  the  new 
axis  is  shifted  by  a  very  small  angle  each  year  from  the  diurnal 
axis  toward  the  line  of  equinoxes.  And  this  line  slides  along 
the  ecliptic,  so  that  the  two  axes  remain  perpetually  at  right 
angles  with  each  other. 

The  rotascope,  a  modification  of  Foucault's  gyroscope,  may 
be  used  to  exhibit  a  very  perfect  illustration  of  the  precession 
of  equinoxes. 

140.  Cause  of  the  slowness  of  precession. — If  the  equatorial 
ring  were  a  separate  body  rotating  about  the  earth  in  its  own 
plane,  its  points  of  intersection  with  the  ecliptic  would  retro- 
grade very  rapidly  by  the  action  of  the  sun  and  moon.     The 
reason  why  the  precession  is  exceedingly  slow  is,  that  while 
the  disturbing  action  is  exerted  only  on  the  ring,  the  force 
around  the  diurnal  axis  consists  of  the  inertia  of  the  entire 
earth.     The  ring  can  not  move  by  itself,  but  must  carry  the 
whole  mass  of  the  earth  with  it. 

141.  The  tropical  and  sidereal  year. — The  fact  of  preces- 
sion shows  that  the  year  has  two  different  values,  according  as 
we  reckon  from  a  star  or  from  an  equinox.     Hence,  the  side- 
real year  is  defined  to  be  the  period  occupied  by  the  sun  in 


84  NUTATION. 

passing  eastward  around  the  heavens  from  a  star  to  the  same 
star  again  ;  and  the  tropical  year,  the  time  of  passing  around 
from  an  equinox  to  the  same  equinox  again  (Art.  86).  As  the 
equinox  moves  westward,  the  sun  reaches  it  sooner  than  if  it 
were  stationary,  and  thus  makes  the  tropical  year  shorter  than 
the  sidereal,  by  the  time  required  to  pass  over  50y,  which  is 
20m.  22.9s.  As  the  tropical  year  is  365d.  5h.  48m.  46.15s. 
(Art.  86),  the  sidereal  year,  therefore,  is  365d.  6h.  9m.  9s. 

Though  the  sidereal  year  is  the  true  period  of  the  earth's 
revolution  about  the  sun,  yet  the  tropical  year  possesses  by  far 
the  greatest  interest,  because  it  is  the  period  in  which  the 
seasons  are  completed. 

142.    Nutation. — By  precession   alone,   the    pole   of  the 
equator  would  move  in  the  circumference  of  a  circle  about  the 
pole  of  the  ecliptic.     But  this  motion  is  modified  by  a  minute 
vibration  from  side  to  side,  as  it 
advances,  so  that  the  line  described  Fig-  46- 

by  the  pole  is  a  delicate  wave  lying     ^.^^^^^'^^^^os,,^ 
along  on  the  circumference,  as  rep-  N\  /M 

resented  in  Fig.  46,  where  P  repre-       \  / 

sents  the  pole  of  the  ecliptic,  and        \  / 

MN  the  path  of  the  pole  of  the         \  / 

equator  around  it.     This  vibratory  \  j 

motion  is  called   nutation.     It  is  \  / 

principally  due  to  the  unequal  ac-  \  / 

tion  of  the  moon  upon  the  equa-  \  / 

torial  ring.  \  / 

The  moon's  action,  at  any  given  \  / 

time,  tends  to  revolve  the  ring  into  \         / 

the   plane   of  its  orbit.     But,   on  \      / 

account  of  the  retrograde  motion  of  \    / 

its  nodes,  the  angle  between  the  \/ 

ring  and  the  moon's  orbit  varies  p 

from  38£°  to  28J°,  going  through  all  the  changes  every  nine- 
teen years.  Owing  to  these  changes  of  position,  the  equinoxes 
will  recede  sometimes  faster,  and  sometimes  slower ;  while  the 
inclination  of  the  equator  to  the  ecliptic  will  also  increase  and 
decrease,  causing  the  poles  of  the  equator  to  oscillate,  as  stated 


ABERRATION    OF    LIGHT.  85 

above.     The  amount,  by  which  the  pole  of  the  equator  moves 
to  and  from  the  pole  of  the  ecliptic  is  IS". 

The  waves  in  the  figure  are  exceedingly  exaggerated.  The 
arc  MN  being  about  TV  of  the  circumference,  the  waves,  if 
truly  represented,  would  be  small  enough  to  cross  the  arc  270 
times. 

143.  Aberration  of  light. — The  heavenly  bodies  suffer  a 
minute  apparent  displacement,  on  account  of  the  progressive 
motion  of  light,  combined  with  the  earth's  motion  in  its  orbit. 
Suppose  the  earth  to  move  from  C  to  E  (Fig. 

47).  while  the  light,  coming  from  S,  describes 

the  line  DE.     If  they  arrive  together  at  the 

point  E,  the  impulse  on  the  retina  of  the  eye 

will  not  be  in  the  same  direction  as  if  the 

observer  had  been  at  rest ;  but  the  light  will 

appear  to  come  in  the  direction  S'E,  the  body 

being  apparently  thrown  forward  from  S  to 

S'.     For,  make  EA  =  DE,  and  complete  the 

parallelogram  CA  ;    and  suppose,  according 

to  the  principle  of  equal  action  and  reaction, 

that  the  light  has  the  motion  I£€  given  to  it, 

in  place  of  the  earth's  motion,  CE  ;  then  the 

two  motions,  EA  and  EC,  will  produce  the  resultant,  EB,  as 

though  the  light  had  come  from  S'  instead  of  S. 

144.  Aberration   illustrated. — The   apparent  direction  of 
any  kind  of  impulse  is  modified  in  the  same  way,  by  the 
motion  of  the  person  who  receives  it.     For  instance,  if  the 
wind  drives  drops  of  rain  in  a  person's  face,  at  a  certain  inclina- 
tion, while  he  is  standing  still,  when  he  comes  to  move  toward 
the  wind,  they  will  strike  him  at  a  less  inclination  with  the 
horizon,  as  though  the  source  of  the  drops  was  further  forward. 
For,  when  the  person  moves,  the  effect  is  the  same  as  if  he 
remained  at  rest,  and  the  wind  were  to  receive  an  increment 
of  velocity  equal  to  his  motion. 

145.  Greatest  and  least  aberration. — The  greatest  aberra- 
tion occurs  when  the  body,  from  which  the  light  comes,  is  in  a 


86  ADVANCE    OF   APSIDES. 

direction  at  right  angles  to  the  line  of  the  earth's  motion. 
The  displacement  is  then  20".5.  When  the  earth  is  moving 
directly  toward  or  directly  from  the  body,  the  aberration  is 
zero.  Therefore,  a  star  in  the  plane  of  the  ecliptic  is  seen  in  its 
true  place  once  every  six  months;  but  three  months  before 
and  three  months  after  either  of  those  times,  it  is  displaced 
20".5  in  opposite  directions,  making  the  total  arc  of  displace- 
ment 41".  But  a  star  at  the  pole  of  the  ecliptic,  being  always 
thrown  forward  of  its  true  place  by  20".5,  will  seem  to  de- 
scribe each  year  a  circle,  whose  diameter  is  41 " .  Between  the 
ecliptic  and  its  poles,  the  apparent  orbit  of  aberration  is  an 
ellipse,  whose  major  axis  is  41 ",  and  whose  minor  axis  increases 
with  the  latitude  of  the  body. 

146.  Velocity  of  light  computed  l>y  aberration. — In   the 
triangle  AEB  (Fig.  47),  AB  represents  the  velocity  of  the 
earth,   AEB  the   observed   aberration,   and  EAB  the   angle 
between  the  line  of  the  earth's  motion  and  the  direction  of 
light.     When  EAB  =  90°.  the  aberration  is  found  to  be  20".5. 
Therefore, 

tan  20".5  :  rad  : :  19  miles  :  192,000 

miles  per  second,  which  is  the  velocity  of  light. 

147.  Advance  of  the  apsides  of  the  earth's  orbit. — It  was 
intimated  in  Art.  74  that  the  Hoe  of  apsides  is  not  stationary. 
If  the  exact  place  of  the  perihelion  among  the  stars  be  noted, 
it  will  be  found  the  next  year  11".5  further  east — that  is,  the 
apsides  advance  11".5  per  year.    But  in  longitude,  the  advance 
is  much  faster,  since  the  vernal  equinox,  from  which  longitude 
is  reckoned,  retrogrades  50£"  per  year.     The  perihelion,  there- 
fore, increases  its  longitude  nearly  62"  each  year. 

As  the  longitude  of  the  perihelion  in  1800  was  279°  30'  8" 
(that  is,  9°  30'  8"  past  the  winter  solstice)  it  must  have  been 
just  at  the  solstice  in  the  year  1247.  For,  9°  30'  8"  ~  6  If"  = 
553  years;  and  1800  —  553  =  1247.  In  a  similar  manner,  it 
is  found  that  the  perihelion  will  be  at  the  summer  solstice  in 
the  year  11741.  In  the  course  of  many  centuries,  the  length 
and  temperature  of  the  seasons  are  modified  by  these  slow 
movements  of  the  equinoxes  and  the  apsides  (Art.  75). 


LONGITUDE   OF  THE   SUN. 


8T 


1  48.  Cause  of  tlie  advance  of  apsides. — The  apsides  of  the 
earth's  orbit  are  made  to  advance  by  the  attraction  of  the 
heavy  planets,  whose  orbits  are  outside  of  it.  The  entire  re- 
sultant of  the  attractions  of  these  planets  upon  the  earth,  is  to 
diminish  a  little  the  earth's  tendency  to  the  sun.  Hence,  as 
the  earth  approaches  one  of  its  apsides,  its  path  is  not  suffi- 
ciently drawn  in  by  the  sun  to  meet  the  former  line  of  apsides 
at  right  angles.  But  it  makes  right  angles  with  a  radius  vec- 
tor a  little  further  on,  which  becomes,  therefore,  the  new  line 
of  apsides. 

1 49.  Sun's  anomaly. — The  sun's  longitude  is  his  distance 
eastward  on  the  ecliptic  from  the  vernal  equinox  (Art.  15). 
Its  aTwmaly  is  its  distance  eastward,  on  the  ecliptic,  from 
perihelion.     The  reason  for  reckoning  motion  from  the  peri- 
helion is,  that  the  angular  velocity  depends  on  it ;  so  that,  to 
find  the  true  longitude  of  the  sun  at  any  time,  we  need  to 
know  how  far  it  is  from  the  perihelion. 

1 5O.  How  to  find  the  true  longitude  of  the  sun  at  a  given 
time. — It  is  first  supposed  that  the  sun  moves  uniformly  in  a 
circle.     And  by  knowing  what  its  mean  motion  is,  and  how 
long  it  is  since  it  passed  the  vernal  equinox,  we  have  its  mean 
longitude  at  once.     But  this  needs  correction  on  account  of 

Fig.  48. 


s' 


the  variable  motion  in  the  ellipse.     Let  E  (Fig.  48)  be  the 
earth ;  PGA,  the  elliptic  orbit  of  the  sun ;  and  BCF,  the  sup- 


88  THE   MOON'S  DISTANCE. 

posed  circular  orbit.  Suppose  the  sun's  mean  place  to  be  at 
S',  and  the  vernal  equinox  at  T  ;  then  its  mean  longitude  is 
TDS',  already  obtained.  The  angle  BES'  is  its  mean  anom- 
aly. But  as  the  sun  has  been  passing  through  the  nearest 
part  of  its  orbit,  its  true  place  is  further  advanced,  as  at  S. 
The  angle  PES  is  the  true  anomaly,  and  the  difference  be- 
tween them — that  is,  S'ES — is  called  the  equation  of  the  center. 
This  equation,  or  correction,  being  found  in  tables  of  the  sun's 
motions,  and  applied  to  the  mean  longitude,  gives  the  true 
longitude. 

If  the  mean  and  true  places  are  considered  as  agreeing  at  P, 
then  the  equation  of  the  center  immediately  becomes  positive, 
and  increases  to  its  maximum  at  C  ;  after  which  it  diminishes, 
and  the  mean  and  true  places  agree  again  at  A.  After  that, 
the  sun  falls  behind  its  mean  place,  and  the  equation  is  neg- 
ative, till  the  sun  reaches  P,  the  greatest  value  being  at  D. 

The  eccentricity  of  the  earth's  orbit  is  so  small,  that  the  sun's 
mean  and  true  places  never  differ  so  much  as  2°,  the  greatest 
equation  of  the  center  being  1°  55'  27". 

151.  The  anomalistic  year. — The  perihelion  is  another 
point  from  which  to  measure  the  revolution  about  the  sun. 
The  time  of  passing  round  from  perihelion  to  perihelion  again 
is  called  the  anomalistic  year.  It  is  4m.  40s.  longer  than  the 
sidereal  year,  or  365d.  6h.  13m.  4:9s. 


CHAPTER  X. 

THE  MOON. — ITS  REVOLUTIONS. — ITS  PHASES. — THE 
CONDITION   OF   ITS  SURFACE. 

152.  Distance  and  dimensions  of  the  moon. — The  moon  is 
a  satellite  of  the  earth,  revolving  about  it  within  a  compara- 
tively small  distance,  and  accompanying  it  in  its  orbit  around 
the  sun.  The  mean  horizontal  parallax  of  the  moon  at  the 


MONTHS.  89 

earth's  equator  being  57'  5",  its  mean  distance  is  found  by 
the  proportion  (Fig.  4), 

sin  57'  5"  :  rad  :  :  3962.8  :  238,650m. 

The  moon's  angular  diameter  is  31'  6"  ;  therefore,  rad  :  sin 
15'  33"  :  :  238,650  :  1080.5  ;  which  is  the  moon's  semi-diameter 
in  miles.  Hence,  the  moon's  diameter  is  2,161  miles. 

The  surfaces  of  the  earth  and  moon  being  as  the  squares  of 
their  radii,  are  as  13  :  1. 

The  volumes  of  the  earth  and  moon  being  as  the  cubes  of 
their  radii,  are  as  49  :  1,  nearly.  But  the  moon's  density  is  so 
small  (3i),  that  the  masses  are  nearly  as  80  :  1. 

The  force  of  gravity  on  the  earth  to  that  on  the  moon  is  as 


(3956f  (T080)"2 


:  :  6  : 


153.  Revolution  about  the  earth.  —  The  slightest  attention 
to  the  position  of  the  moon,  from  night  to  night,  shows  that  it 
moves  eastward,  among  the  stars,  several  degrees  every  day. 
If  the  instruments  of  the  observatory  be  employed  to  measure 
its  right  ascension  and  declination,  as  in  the  case  of  the  sun 
(Arts.  58,  59),  it  is  ascertained  that  the  moon  describes  nearly 
a  great  circle,  inclined  about  5°  to  the  ecliptic,  and  occupies 
27.32  days  in  returning  to  the  same  place  among  the  stars. 

The  inclination  of  the  moon's  orbit  to  the  ecliptic  va- 
ries from  5°  20'  6"  to  4°  57'  22"  ;  but  its  mean  value  is  5° 
8'  55". 

154.  Months.  —  The  period  just  mentioned,  in  which  the 
moon  makes  a  revolution  from  a  star  to  the  same  star  again,  is 
called  the  sidereal  month.     The  time  occupied  in  making  a 
revolution  relatively  to  the  sun,  instead  of  a  star,  is  called  a 
synodical  month.     This  is  more  than  two  days  longer  than  the 
sidereal  month  ;   for  the  moon's  daily  progress  is  about  13°  ; 
and  during  the  27  days  of  its  revolution,  the  sun,  at  the  rate  of 
1°  per  day,  will  advance  27°,  requiring  more  than  two  addi- 
tional days  for  the  moon  to  overtake  it. 

The  mean  length  of  the  synodical  month  is  29.53  days. 


90  MOON'S  ORBIT. 

155.  Nodes. — The  points  where  the  moon's  path  cuts  the 
circle  of  the  ecliptic  are  called  the  moon's  nodes.     The  ascend- 
ing node  is  the  one  through  which  the  moon  passes  from  the 
south  to  the  north  side  of  the  ecliptic  ;  the  other,  180°  from  it, 
is  called  the  descending  node. 

156.  The  moon's  positions  in  relation  to  the  sun. — The 
moon  is  said  to  be  in  conjunction  with  the  sun,  when  both 
bodies  have  the  same  longitude;    in  opposition,  when  their 
longitudes  differ  by  180°.-    The  conjunction  and  opposition 
are  called  by  the  common  name  of  syzygies. 

When  the  longitude  of  the  moon  is  90°,  or  270°  greater 
than  that  of  the  sun,  it  is  said  to  be  in  quadrature. 

The  points  midway  between  syzygies  and  quadratures  are 
called  octants. 

The  period  in  which  the  moon  passes  from  any  one  of  these 
points  to  the  same  point  again — that  is,  a  synodical  month — is 
also  called  a  lunation. 

157.  To  find  the  synodical  month. — The  synodical  month 
is  best  obtained  by  comparing  ancient  and  modern  eclipses. 
An  eclipse  of  the  sun  takes  place  at  the  time  of  conjunction. 
If  then,  the  whole  interval  between  the  recorded  date  of  a 
solar  eclipse,  which  occurred  before  the  Christian  era,  and  the 
time  of  another,  which  occurred  recently,  be  divided  by  the 
number  of  intervening  lunations,  the  quotient  is  a  very  accu- 
rate expression  of  the  mean  synodical  month. 

The  mean  synodical  month,  as  thus  obtained,  is  29d.  12h. 
44m.  3s.  =  29.5309  days. 

158.  To  find    the   sidereal    month.— Dividing    360°   by 
365.25635,  the  number  of  days  in  a  sidereal  year,  we  have 
0°.9856,  the  mean  daily  progress  of  the  sun.     Multiplying  this 
by  29.53,  the  number  of  days  in  a  synodical  month,  we  find 
29°.105,  the  arc  passed  over  by  the  sun  in  that  time.    Now, 
the  moon  passes  over  360°  4-  29°.105  in   a  synodical  month, 
but  only  360°  in  a  sidereal  month.     Hence,  we  have  the  pro- 
portion, 360°  +  29°.  105  :  360°  ::  29.53d.  :  L>7.32d. 

The  sidereal  month,  more  exactly,  is  27d.  7h.  43rn.  lis. 


LIBRATION   IN  LONGITUDE.  91 

159.  Form  of  the  moon's  orbit. — It  is  ascertained  by  the 
same  method  as  was  described  (Art.  71),  that  the  moon's  orbit 
is  an  ellipse,  one  of  whose  foci  is  at  the  earth.     The  moon's 
apparent  diameter  varies  from  33'  32"  to  28'  48".     Therefore, 
the  greatest  and  least  distances  of  the  moon  from  the  earth  are 
in  the  ratio  of  these  numbers,  or  as  7  :  6,  nearly ;  and  the  ec- 
centricity =  T]3  or  0.076,  which  is  about  4|  times  as  great  as 
the  eccentricity  of  the  earth's  orbit  (Art  73).     Yet  a  figure  in 
the  exact  form  of  the  moon's  orbit  could  not  be  distinguished 
from  a  circle,  since  the  major  axis  would  exceed  the  minor  by 
less  than  j-^  of  its  length. 

The  point  of  the  moon's  orbit  nearest  the  earth  is  called  the 
perigee,  the  most  distant  point  the  apogee. 

160.  The  moon's  diurnal  motion. — The  moon  not  only  re- 
volves about  the  earth,  but  also  on  its  own  axis  in  the  same 
length  of  time — that  is,  once  in  27.32  days ;   and  its  axis  is 
nearly  perpendicular  to  the  plane  of  its  orbit.     This  rotation 
is  indicated  by  the  fact  that  the  same  side  of  the  moon  is  al- 
ways presented  toward  the  earth.     If  it  should  pass  around  the 
earth,  and  not  turn  upon  an  axis,  it  would  obviously  present 
all  sides  to  us  in  the  course  of  each  revolution. 

But  though  it  keeps  the  same  side  toward  the  earth,  it  pre- 
sents all  sides  to  the  sun  once  in  each  synodical  month  ;  there- 
fore, the  days  and  nights  on  the  moon  are  nearly  30  (29.53) 
times  the  length  of  those  on  the  earth. 

161.  The  moon's  librations. — Though  the  same  side  of  the 
moon  is  turned  toward  us  on  the  whole,  yet  there  are  slight 
apparent  oscillations,  by  which  narrow  portions  of  the  other 
hemisphere  alternately   come  into  view.      These  are   called 
librations.     They  are  of  three  kinds :  the  libration  in  longi- 
tude, the  libration  in  latitude,  and  the  diurnal  libration. 

162.  The  libration  in  longitude. — By  this  libration  we  ex- 
tend our  view  a  little  further  round  upon  the  moon's  equator, 
first  on  one  side,  then  on  the  other,  every  sidereal  month. 

It  arises  from  the  fact  that  while  the  moon  rotates  uni- 
formly on  its  axis,  it  revolves  in  its  elliptical  orbit  with  un- 


92  REVOLUTION  ABOUT  THE   SUN. 

equal  angular  velocity.  Near  the  apogee,  where  it  moves 
slowest,  it  rotates  more  than  90°  on  its  axis,  while  passing  just 
90°  around  us,  and  thus  reveals  a  little  of  the  remote  hemi- 
sphere on  the  eastern  side.  Near  the  perigee,  on  the  other 
hand,  where  the  orbit  motion  is  rapid,  it  makes  less  than  one- 
fourth  of  a  rotation,  while  going  90°  around  the  earth.  This 
brings  into  view  a  little  of  the  other  hemisphere  on  the  western 
limb. 

If  the  moon's  orbit  were  a  circle,  there  would  be  no  libra- 
tion  of  longitude. 

163.  The  libration  in  latitude. — As  the  name  implies,  this 
libration  extends  our  view  alternately  north  and  south  on  the 
moon's  meridian.     As  the  moon's  equator  is  a  little  inclined 
to  the  plane  of  its  orbit,  its  north  and  south  poles  are  brought 
alternately  toward  us,  just  as  the  earth's  poles  are  presented  in 
turn  toward  the  sun  every  year.     The  mean  value  of  the  incli- 
nation of  the  moon's  equator  to  its  orbit  is  6°  39'. 

If  the  moon's  equator  and  itsarbit  were  in  the  same  plane, 
there  would  be  no  libration  of  latitude. 

164.  The  diurnal  libration. — This  is  the  effect  of  diurnal 
parallax.     When  the  moon  is  on  the  meridian,  we  view  it 
nearly  as  from  the  center  of  the  earth ;  but  when  it  is  at  the 
horizon,  we  see  it,  as  it  were,  from  a  position  near  4,000  miles 
higher,  and  extend  our  vision  a  little  distance  over  its  western 
limb  at  rising,  and  its  eastern  at  setting. 

165.  Apparent  diameter  on  the  meridian  and  at  the  hori- 
zon.— The   distance  of  the  moon  from  the  earth  is  about  60 
times  the  radius  of  the  earth.     Therefore,  when  the  moon  is  on 
the  meridian,  as  it  is  -£-$  nearer  than  when  at  the  horizon,  its 
apparent  diameter  is  -^  greater.     This  change,  equal  to  about 
30",  is  too  small  to  be  perceived  by  the  eye,  but  can  be  meas- 
ured by  instruments. 

166.  The  mooris  revolution  about  the  sun. — While   the 
moon  revolves   about  the  earth,  the  earth  revolves  about  the 
sun,  at  a  distance  400  times  as  great.     For,  238,650  x  400  = 
95,460,000. 


WHAT  FORCES   CONTROL   THE   MOON.  93 

Therefore,  the  moon  really  has  a  third  revolution  —  namely, 
that  in  company  with  the  earth  around  the  sun.  And  this  is 
far  greater  than  its  other  revolutions,  which  have  been  de- 
scribed. A  point  of  the  moon's  equator,  in  its  diurnal  motion, 
goes  only  10  miles  per  hour.  Around  the  earth,  the  moon's 
velocity  is  nearly  2,300  miles  per  hour  ;  but  around  the  sun,  it 
is  more  than  68,000  miles  per  hour. 

167.  Form  of  path  around  the  sun.  —  Whenever  a  body  re- 
volves about  a  center,  while  that  center  is  itself  in  motion,  the 
body  describes  a  species  of  curve,  called  an  epicycloid.  The 
moon's  path  about  the  sun  is  a  waving  epicycloid.  Let  the 
small  circles  at  A,  B,  etc.  (Fig.  49),  represent  the  size  of  the 


Fig. 


moon's  orbit,  and  let  AE  be  an  arc  of  the  earth's  orbit,  the 
sun  being  at  the  intersection  of  the  dotted  lines  when  pro- 
duced. While  the  moon  describes  one  half  of  its  orbit,  the 
earth  goes  over  ^  of  its  annual  circuit — that  is,  from  A  to  E. 
Therefore,  the  earth  being  at  A,  suppose  the  moon,  in  quadra- 
ture on  the  left,  beginning  to  describe  the  semicircle  nearest  the 
sun.  When  the  earth  reaches  B,  the  moon  has  passed  to  the 
octant  m  •  at  C,  the  moon  is  in  conjunction  ;  at  D,  it  is  at  the 
next  octant ;  and  at  E,  it  is  again  in  quadrature  on  the  right, 
having  described  a  semicircle  relatively  to  the  earth.  But,  in 
relation  to  the  sun,  it  has  passed  over  the  curve  inside  of  the 
earth's  path,  from  A  to  E.  At  E,  it  crosses  the  earth's  path, 
and  while  describing  the  outer  semicircle,  it  advances  with  the 
earth  a  distance  equal  to  AE,  on  the  outside.  Thus,  the 
moon's  path  around  the  sun  consists  of  25  undulations,  so 
slight  that,  if  represented  alone,  the  whole  would  scarcely  be 
distinguished  from  the  earth's  orbit. 

168.  By  what  forces  the  moon  is  mainly  controlled. — Since 
the  moon  describes  around  the  sun  an  orbit  at  the  mean  dis- 


94  MOO^S   PHASES. 


tance*  of  the  earth's  orbit,  and  in  the  same  time,  it  must  be 
subject  to  the  same  projectile  and  centripetal  forces.  If  the 
earth,  therefore,  were  to  be  annihilated,  the  moon's  path  about 
the  sun  would  not  be  essentially  disturbed  ;  the  waves  only 
would  cease,  and  the  orbit  become  an  exact  ellipse. 

The  relative  attractions  of  the  earth  and  sun,  r        -1  on 
the  moon,  are  estimated  by  the  formula  proved  in  *Trt:  92, 

c  oo  -.     Considering  the  radius  of  the  moon's  orbit  =  1,  that 
t 

of  the  earth's  orbit  is  about  400  ;  and  the  times  are  27.32d.  and 
365.  25d.,  respectively.  Hence,  attraction  to  the  earth  :  that 

1  400 

to  the  sun  :  :      -        :  :  :  1  :  2.2,  nearly.    Therefore, 


the  sun,  though  so  very  far  from  the  moon,  exerts  upon  it 
2J  times  more  attraction  than  the  earth  does. 

169.  How  the  earth's  action  causes  the  waves  in  the  moon's 
path.  —  When  the  moon  is  in  conjunction,  as  at  C,  the  earth 
draws  it  away  from  the  sun,  so  that  it  begins  to  move  further 
off,  as  at  D,  E,  etc.,  till  it  reaches  opposition.     But,  at  opposi- 
tion, the  earth  is  on  the  same  side  as  the  sun,  and  increases  the 
moon's  tendency  toward  it,  so  that  the  moon  begins  to  move 
toward  the  sun,  and  continues  approaching  till  it  reaches  con- 
junction again.     But,  in  describing  the  wave  line,  the  moon 
sometimes  gets  in  advance  of  the  earth  in  its  orbit,  as  at  A, 
and  then  falls  behind,  as  at  E.     For,  the  earth  at  A  draws  the 
moon  backward,  and  it  falls  further  and  further  back,  till  it  is 
behind  the  earth  in  its  motion,  as  at  E,  where  the  earth,  having 
overcome  the  backward  motion,  draws  it  forward,  till  it  passes 
by,  and  is  again  in  advance  of  the  earth.     Thus,  in  the  moon's 
great  revolution  around  the  sun,  we  may  regard  its  path  as 
thrown  into  the  waving  line  by  the  small  disturbing  influences 
of  the  earth. 

170.  Phases  of  the  moon.  —  The  moon  is  not  self-luminous, 
and  is  seen  only  as  it  reflects  to  us  the  light  which  falls  upon 
it.     The  several  forms  which  the  part  illuminated  by  the  sun 
presents  to  our  view,  are  called  phases. 

The  circle  of  illumination^  or  the  terminator  ',  is  the  circle 


MOON'S  PHASES.  95 

which  separates  the  hemisphere  enlightened  by  the  sun  from 
the  dark  hemisphere,  and  is  perpendicular  to  the  sun's  rays 
which  fall  on  the  moon.  The  circle  of  the  disk  is  that  which 
separates  the  hemisphere  turned  toward  the  earth  from  the  op- 
posite one,  and  is  perpendicular  to  our  line  of  vision.  The 
phase  depends  on  the  size  of  the  angle  formed  at  the  moon, 
between  the  solar  ray  and  our  visual  line. 


Fig.  50. 

D 

tf^c               I1 

_D^-^^y^-x_    * 

n 

Let  the  earth  be  at  E  (Fig.  50),  and  the  moon  in  several  po- 
sitions, A,  B,  etc.,  and  let  the  lines  AS,  BS,  etc.,  be  directed 
toward  the  sun.  At  A,  the  moon  is  in  conjunction,  and  wholly 
invisible — this  is  called  new  moon;  and  the  angle  SAE,  be- 
tween the  solar  ray  and  visual  ray,  is  180°.  From  A  to  C 
(as  at  B),  the  phase  is  called  crescent ;  and  the  angle,  SBE,  is 
obtuse.  The  first  quarter  occurs  at  C,  the  quadrature,  where 
SCE  is  a  right  angle.  From  C  to  F  (as  at  D),  the  phase  is 
called  gibbous  ;  in  this  phase,  the  angle,  SDE,  is  always  acute. 
At  F,  the  moon  is  in  opposition,  and  wholly  illuminated.  This 
is  called  full  moon  •  the  angle,  SFE,  is  0°.  From  F  to  A, 
the  phases  are  repeated  in  reverse  order,  the  last  quarter  being 
at  H.  The  outer  figures  at  B,  C,  etc.,  show  the  corresponding 
phase. 


96  INEQUALITIES   OF   THE   MOON'S   SUKFACE. 

171.  The  meridian  altitudes  of  the  moon  at  a  given  phase. 
— It  is  generally  observed  that  at  a  given  age  of  the  rnoon,  for 
instance  at  the  full,  its  meridian  altitude  is  very  different  at 
different  seasons  of  the  year.     This  is  readily  explained,  by 
noticing  the  moon's  relations  to  the  sun.     As  the  moon's  path 
is  everywhere  near  the  ecliptic,  the  new  moon  will  culminate 
at  a  high  point  when  the  sun  does — that  is,  in  the  summer. 
But,  in  the  same  season,  the  fall  moon,  being  opposite  to  the 
sun,  will  culminate  low.     On  the  contrary,  when  the  sun  is  in 
the  most  southern  part  of  the  ecliptic,  and  culminates  low,  as 
is  the  case  in  winter,  the  new  moon  will  do  so  likewise ;  but 
the  full  moon  will  culminate  at  a  high  point.     In  the  tropical 
winter,  therefore,  when  the  sun  is  absent  for  months,  the  moon, 
whenever  near  the  full,   circulates  round    the   sky  without 
setting. 

172.  The  harvest  moon. — This  name  is  given  to  the  full 
moon  which  occurs  nearest  to  the  autumnal  equinox,  Septem- 
ber 22d,  and  which  rises  from  evening  to  evening  with  a  less  in- 
terval of  time  than  the  full  moon  of  any  other  season. 

The  sun  being  at  the  autumnal  equinox,  the  moon  is  near 
the  vernal  equinox,  and  at  sunset,  the  southern  half  of  the 
ecliptic  is  above  the  horizon,  and  makes  the  smallest  possible 
angle  with  it.  It  is  this  small  angle,  made  by  the  ecliptic,  and 
therefore  by  the  moon's  orbit  with  the  horizon,  which  causes 
the  small  interval  in  the  time  of  the  moon's  rising  from  one 
evening  to  another ;  for,  as  the  moon  advances  13°  each  day 
in  its  orbit,  this  arc  is  so  oblique  to  the  horizon  that  its  two 
extremities  rise  with  only  a  few  minutes'  difference  of  time ; 
but  ih& place  of  rising  moves  rapidly  northward. 

The  harvest  moon  attracts  most  attention  in  high  latitudes, 
where  the  angle  between  the  ecliptic  and  horizon  is  smaller, 
and  therefore  the  intervals  of  time  are  less. 

The  moon  passes  the  vernal  equinox  every  month,  and 
therefore  rises  with  the  same  small  intervals.  But  when  the 
moon  is  not  full  at  the  same  time,  the  circumstance  is  un- 
noticed. 

173.  Inequalities  of  the  moorfs  surface. — These  are  clearly 


FORMS   OF  VALLEYS.  97 

revealed  by  the  changing  direction  of  the  sun's  rays.  As 
the  terminator  advances  over  the  disk,  the  light  strikes  the 
highest  peaks,  which  appear  as  bright  points  a  little  way  upon 
the  dark  part  of  the  moon.  After  the  terminator  has  passed 
over  them,  they  project  shadows  away  from  the  sun,  which 
correspond  to  the  apparent  shape  of  the  elevations,  and  grow 
shorter  as  the  rays  fall  more  nearly  vertical.  And  again,  in 
the  waning  of  the  moon,  the  shadows  are  cast  in  the  opposite 
direction,  lengthening  until  the  dark  part  of  the  disk  reaches 
them,  and  the  summits  once  more  become  isolated  bright 
points,  and  then  disappear.  Fig.  2,  Frontispiece,  will  give 
some  idea  of  these  appearances. 

174.  Forms  of  valleys. — The  most  striking  characteristic 
of  the  moon's  surface  is  its  numerous  circular  valleys.  A  few 
are  represented  in  Figs.  1  and  2,  Fr.  The  smaller  and  more 
regular  ones  are  of  all  sizes,  from  one  or  two  miles  in  diameter 
up  to  sixty  miles.  These  are  numbered  by  hundreds.  The 
mountain  ridge  which  surrounds  one  of  these  cavities  is  a  ring, 
very  steep  and  precipitous  on  the  inner  side  ;  but  externally  it 
falls  off  by  a  rugged  but  gradual  slope.  These  ridges  are 
called  ring-mountains.  In  the  central  part  of  the  cavity  are 
generally  one  or  more  steep,  conical  mountains.  Some  of  the 
principal  ring-mountains  are  No.  1.  Tycho ;  2.  Kepler ;  3.  Co- 
pernicus;  etc.  (Fig.  1,  Fr.) 

There  is  another  class  of  larger  but  less  regular  cavities, 
sometimes  called  bulwark  plains.  Their  diameters  are  often 
more  than  one  hundred  miles.  These  are  also  surround- 
ed by  rough  mountain  masses  arranged  in  a  circle.  Over 
these  plains  are  sparsely  scattered  small  conical  and  ring  moun- 
tains. 

There  are  still  larger  tracts,  more  level  than  the  general 
lunar  surface,  and  of  a  darkish  hue,  which  still  retain  the  name 
of  seas,  formerly  given  them,  though  they  are  covered  with 
permanent  inequalities,  and  show  no  signs  of  being  fluid.  Ex- 
amples of  these  are:  A.,  mare  humorum;  B,  mare  nubium, 
etc.  (Fig.  1,  Fr.) 

Besides  the  ridges  of  mountains  inclosing  the  circular  vaL- 

7 


98  LUNAR  MOUNTAINS. 

leys,  there  are  occasional  chains  and  spurs,  having  more  resem- 
blance to  terrestrial  ranges.* 

175.  Luminous  radiations. — At  full  moon,   all   shadows 
disappear,  because  the  light  falls  in  the  direction  of  our  line  of 
vision.     But  at  that  time  another  peculiarity  presents  itself. 
From  a  few  of  the  large  ring-mountains  there  radiate  a  great 
number  of  luminous  stripes,  nearly  in  straight  lines,  and  ex- 
tending, in   some   cases,   hundreds   of  miles.     They   are  not 
ridges,  as  they  cast  no  shadows  when  the  terminator  passes 
them ;  and  the  difference  of  illumination  must  result  from  the 
different  nature  of  their  material.     They  are  sometimes  called 
lava-lines.     The  most  extensive  system  occurs  around  Tycho, 
marked  1,  in  Fig.  1,  Fr. 

176.  Surface  rigid  and  angular. — Every  part  of  the  moon's 
surface  has  the  appearance  of  rocky  hardness.     The  interior 
slopes  of  the  ring-mountains  are  steep,  rough,  and  angular. 
The  conical  peaks  within  them  appear  like  isolated  rocks,  re- 
sembling the  needles  of  the  Alps.     The  surface  nowhere  gives 
indication  of  having  been  softened  down  by  the   action   of 
water. 

177.  Probable  volcanic  origin. — The  circular  cavities,  with 
steep  and  rugged  sides,  appear  like  vast  craters,  and  the  moun- 
tains within  them  like  volcanic  cones,  more  recently  thrown 
up.     Nearly  every  part  of  the  hemisphere  presented  to  our 
view  exhibits  these  indications  of  former  volcanic  action,  on  a 
scale  far  beyond  any  thing  on  the  earth.     But  there  is  no  evi- 
dence of  volcanic  action  at  present. 

178.  Height  of  lunar  mountains. — One  method  of  measur- 
ing the  height  of  a  lunar  mountain  is  the  following.     Let  the 
light  from  the  sun,  S  (Fig.  51),  pass  the  moon's  surface  at  O,  and 
illuminate  the  summit  of  the  mountain,  MF.    The  observer,  E, 
at  the  earth,  sees  M  as  a  bright  point,  on  the  dark  portion  be- 

*  The  lunar  map  of  Beer  and  Madler,  2|  feet  in  diameter,  contains  a  very 
perfect  delineation  of  the  mountains  and  valleys  of  the  moon,  accompanied  "by 
their  names. 


NO   ATMOSPHERE   OE   YAPOE. 


99 


yond  the  terminator  O.  With  a  micrometer,  he  measures  the 
angle  OEM,  subtended  by  OM  ;  and  by  observing  the  elon- 
gation of  this  point  from  the  sun,  MES,  he  obtains  EMS. 
By  these  angles,  and  EM,  the  known  distance,  OM  is  com- 
puted. Then,  in  the  right-angled  triangle  OMC,  OC  and 
OM  furnish  CM,  from  which  CF  being  subtracted,  the  height 
of  the  mountain  remains. 

Fig.  51. 


The  height  of  a  mountain  may  also  be  determined  by  meas- 
uring the  length  of  its  shadow,  and  the  inclination  of  the  solar 
ray  which  casts  it. 

The  highest  of  the  lunar  mountains  have  an  elevation  of  4£ 
miles,  and  great  numbers  of  them  exceed  three  miles.  Thus, 
the  mountains  of  the  moon  are  proportionally  inuch  greater 
than  those  of  the  earth.  For,  while  the  diameter  of  the  moon 
is  not  much  more  than  one-fourth  as  great  as  the  earth's  diam- 
eter, its  mountains  are  about  equal  in  height  to  the  mountains 
on  the  earth. 

179.  No  atmosphere  or  vapor. — If  any  kind  of  atmosphere 
were  spread  over  the  disk  of  the  moon,  it  would  reflect  the 
sun's  light  so  strongly  as  to  dim  the  features  of  the  solid  sur- 
face. Nothing  of  the  kind  is  ever  perceived.  No  terrestrial 


100  NO   ATMOSPHERE   OR  VAPOR. 

objects,  however  near,  ever  exhibit  greater  sharpness  of  outline 
than  the  inequalities  of  the  moon  ;  and  they  never  vary'  in  this 
respect,  except  in  a  manner  which  is  obviously  occasioned  by 
our  own  atmosphere. 

But  the  severest  test  of  a  perceptible  atmosphere  would  be 
the  effect  on  a  star  at  the  beginning  and  end  of  its  occultation 
by  the  moon.  Let  AB  (Fig.  52)  be  the  edge  of  the  moon's 

Fig.  52. 


disk,  and  CD  that  of  the  atmosphere  around  it.  The  light 
from  the  star  S  will,  according  to  the  laws  of  optics,  be  re- 
fracted toward  the  moon  in  entering  its  atmosphere,  and  as 
much  more  in  the  same  direction  in  leaving  it ;  so  that  it  will 
reach  the  observer  at  E,  appearing  to  come  from  S',  when  the 
star  is  really  behind  the  moon  at  S.  Thus,  it  will  appear  to  be 
detained  in  its  diurnal  motion  as  it  approaches  the  edge  of  the 
moon,  and  to  arrive  only  to  S'  when  it  has  really  reached  the 
position  S.  So,  also,  in  reappearing  at  the  opposite  limb,  the 
star  will  seem  to  have  advanced  to  the  edge,  when  it  is 
still  behind  the  moon ;  so  that,  after  coming  into  view,  and 
before  passing  by  the  atmosphere,  it  will  again  appear  to  be 
detained  in  its  diurnal  motion.  Since  it  disappears  too  late, 
and  reappears  too  early,  the  duration  of  occultation  is  too 
short. 

Besides  this  irregularity  in  its  motion,  its  brightness  will 
also  be  a  little  dimmed  by  the  obstruction  of  the  atmosphere, 
just  before  disappearing,  and  just  after  reappearing. 

Now,  the  nicest  observations  have  failed  to  show  either  of 
these  effects.  The  diurnal  motion  is  uniform  up  to  the  very 
edge  of  the  disk,  and  the  actual  continuance  of  occultation  is 
equal  to  the  calculated  duration.  And,  as  to  loss  of  light, 
the  star  at  its  full  brightness  disappears  all  at  once,  with  a 
suddenness  which  is  startling.  Its  reappearance  is  equally 
sudden,  and  without  any  change  of  intensity  in  its  light.  The 
moon,  therefore,  has  no  appreciable  atmosphere. 


I 

VIEW   OF   EARTH   FROM   MOON.  101 

180.  Changes  of  temperature  on  the  moon. — The  moon's 
equator  makes  an  angle  of  only   1J°  with  the  ecliptic,  and 
therefore  experiences  no  perceptible  change  of  seasons ;  but  its 
diurnal  rotation  is  so  slow,  that  the  extremes  of  heat  and  cold 
during  each  day  are  excessive.     A  place  on  the  moon  is  ex- 
posed to  the  full  power  of  the  sun's  rays  for  about  two  weeks, 
and  then  is  for  as  long  a  time  turned  away  from  the  sun,  with- 
out clouds,  or  even  air,  to  prevent  the  free  radiation  of  heat. 

181.  View  of  the  earth  from  the  moon. — 

1.  As  to  magnitude. — The  apparent  dimensions  of  the  two 
bodies,  as  seen  one  from  the  other,  are  proportional  to  their 
real  dimensions.     Hence,  in  diameter,  the  earth  as  seen  from 
the  moon  is  3f  times  as  large  as  the  moon  viewed  from  the 
earth,  and  in  area  is  about  13  times  as  large. 

2.  As  to  phase. — It  is  obvious,  from  Fig.  50,  that  when  the 
full  moon  is  presented  to  the  earth,  the  earth's  dark  side  is 
toward  the  moon,  and  the  reverse.     Also,  that  when  we  see 
the  gibbous  phases  of  the  moon,  a  spectator  on  the  moon  would 
see  crescent  phases  of  the  earth ;  for  the  angle  SED  or  SEG 
would  then  be  obtuse.     In  like  manner,  the  relative  phases  are 
in  every  case  supplementary  to  each  other.     This  relation  ex- 
plains the  well-known  fact  that  near  the  time  of  new  moon, 
the  part  of  the  moon  not  directly  enlightened  by  the  sun  is 
distinctly  visible.     It   is  then   illuminated   indirectly  by  the 
earth,  which  is  nearly  full  as  seen  from  the  moon,  and  reflects 
a  strong  light  upon  it. 

For  the  same  reason,  the  moon  can  be  faintly  seen  in  a  total 
solar  eclipse. 

3.  As  to  position  in  the  sky. — The  earth  seen  from  the  moon 
has  no  apparent  diurnal  rotation,  as  all  other  heavenly  bodies 
have,  but  remains  nearly  fixed  in  the  same  part  of  the  sky. 
This  is  owing  to  the  fact  that  the  moon's  monthly  motion  and 
its  diurnal  motion  are  at  the  same  rate  in  the  same  direction, 
so  that  one  apparent  motion  of  the  earth  neutralizes  the  other. 
Hence,  a  spectator  occupying  the  middle  of  the  moon's  disk 
sees  the  earth  perpetually  near  his  zenith.     Another,  at  the 
edge  of  the  disk,  sees  it  always  near  the  same  point  of  the 
horizon. 


102  MOON'S   GRAVITY  DISTURBED. 

The  first  and  second  librations  of  the  moon,  since  they  vary 
the  spectator's  position  a  little  in  relation  to  the  disk,  merely 
cause  small  oscillations  of  the  earth's  place  in  the  sky. 

4.  As  to  surface. — The  earth,  by  its  rotation,  presents  all  its 
parts  to  the  view  of  the  nearer  hemisphere  of  the  moon  once  in 
25  hours.  To  the  other  hemisphere  it  never  appears  at  all. 

On  account  of  its  nearness,  and  its  great  size,  we  might  sup- 
pose that  the  geographical  features  of  the  earth  would  be 
very  conspicuous  to  a  spectator  on  the  moon,  and  that  the 
nature  of  its  surface  in  nearly  all  respects  could  be  thoroughly 
observed.  But  the  deep  and  dense  atmosphere  of  the  earth 
would  reflect  an  intense  light,  so  as  probably  to  render  the  in- 
equalities of  the  terrestrial  surface  nearly  invisible ;  and  when- 
ever clouds  prevail  over  a  country,  that  portion  of  the  earth's 
disk  would,  of  course,  be  entirely  hidden  from  view. 


CHAPTER  XL 

DISTURBANCES  OF  THE   MOON'S  MOTION  CAUSED   BY  THE  SUN. 

182.  Why  the  sun  disturbs  the  moon7 s  revolutions  around 
the  earth. — If  the  sun  were  at  an  infinite  distance  from  the 
earth  and  moon,  however  great  its   attraction  might   be,  it 
would  not  disturb  their  mutual  relations,  because  it  would  act 
on  both  exactly  alike.     Though  the  sun's  distance  from  them 
is  very  great,  being  400  times  their  distance  from  each  other, 
yet  .the  difference  of  action  is  sufficient  to  produce  sensible  dis- 
turbances.    These  disturbances  are  caused  in  part  by  difference 
of  distance,  and  in  part  by  difference  of  direction. 

183.  The  moon's  gravity  diminished  at  syzygies,  and  in- 
creased at  quadratures. — When  the  moon  is  in  conjunction, 
the  sun  attracts  it  more  than  it  does  the  earth,  in  the  ratio  of 
4002  :  39 92,  and  thus  diminishes  the  moon's  tendency  to  the 
earth.     In  opposition,  the  sun  attracts  the  moon  less  than  it 
does  the  earth,  in  nearly  the  samp  ratio,  which,  as  before,  di- 


THE   SUX  S  DISTURBING  EFFECT. 


103 


minislics  the  moon's  tendency  to  the  earth.  Therefore,  at  the 
syzygies,  tho  moon's  gravity  to  the  earth  is  diminished.  And 
the  diminution  is  computed  to  be  about  ^  of  the  whole. 

In  quadrature,  the  sun  attracts  the  moon  in  a  line  slightly 
oblique  to  that  in  which  it  attracts  the  earth.  Hence,  there  is 
a  sir. all  component  of  its  action  directed  toward  the  earth. 
Therefore,  at  the  quadratures,  the  moon's  gravity  to  the  earth 
is  increased.  This  increase  is  proved  to  be  about  T-i^  of  the 
whole,  or  one-half  as  great  as  the  diminution  at  syzygies. 

As  the  diminution  at  syzygies  is  more  than  the  increase  at 
quadratures,  the  entire  effect  of  the  sun's  influence  is  to  dimin- 
ish the  moon's  gravity  to  the  earth,  and  thus  cause  it  to  revolve 
in  a  larger  orbit  than  it  would  do  if  the  sun  did  not  exist. 
The  moon's  gravity  to  the  earth  is  diminished  by  3-^,  in  con- 
sequence of  the  sun's  action. 


184.  The  sun's  disturbing  effect  repre- 
sented geometrically.  —  Let  ABCD  (Fig. 
53)  be  the  moon's  orbit  described  about 
the  earth  E,  and  S  the  place  of  the  sun. 
Suppose  the  moon  at  M.  Let  ES  rep- 
resent the  attraction  of  the  sun  upon  the 
earth.  Then  (Art.  128),  SM2  :  SE2  :  :  SE  : 

3 

=  the  attraction  of  the  sun  upon  M, 


QT?3 


in  the  direction  MS.    Make  MG-  = 


draw  MF  equal  and  parallel  to  ES,  and 
complete  the  parallelogram  MFGH. 
Resolve  the  force  MG  into  MF  and  MH. 
Since  the  component  MF  is  equal  and 
parallel  to  ES,  which  is  the  sun's  attrac- 
tion on  the  earth,  it  produces  no  disturb- 
ance ;  and  the  only  force  which  can  dis- 
turb the  relations  of  M  and  E  is  the 
other  component  MH.  This  line  lies 
in  various  positions,  and  is  of  various 
lengths,  according  to  the  place  of  M.  It 
is  convenient  to  reduce  it  to  two  ele- 


104: 


EVECTION. 


ments  called  the  radial  and  the  tangential  disturbing  forces. 
Draw  MO  tangent  to  the  orbit,  and  EM  joining  the  earth 
and  moon ;  then,  MET  may  be  resolved  into  the  radial  force 
MP,  increasing  or  diminishing  the  moon's  gravity  to  the  earth, 
and  the  tangential  force  MO,  which  increases  or  diminishes 
the  velocity  of  the  moon.  In  the  figure,  the  position  of  MH 
is  such,  that  MP  increases  the  gravity,  and  MO  accelerates. 

Near  the  quadratures,  MP  acts  toward  E ;  and  near  the 
syzygies,  it  acts  away  from  E.  MO  accelerates  on  the  quad- 
rants DA  and  BC,  and  retards  on  AB  and  CD.  , 

185.  Equations  for    correcting   the   moon's  place. — The 
moon's  path  being  elliptical,  and  its  motion  being  subject  to 
several  disturbances,  its  true  longitude  for  a  given  time  can 
not  be  found,  except  by  applying  various  corrections. 

186.  The  equation  of  the  center. — First  suppose  the  moon 
to  revolve  uniformly  in  a  circular  orbit,  and  then,  as  in  the 
case  of  the  sun  (Art.  150),  apply  the  equation  of  the  center  to 
change  its  place  for  the  variable  motion  in  the  ellipse.     The 
moon's  orbit  being  more  eccentric  than  the  earth's,  its  great- 
est equation  of  the  center  is  6°  IT  13",  while  the  sun's  is  less 
than  2°. 


187.  Evection. — A  cor- 
rection must  be  applied  on 
account  of  the  change  of  ec- 
centricity caused  by  the 
sun's  disturbance.  This 
change  of  eccentricity  is 
called  evection.  It  is  caused 
by  the  radial  disturbance  r> 


Fig 


.54. 

C 


-B 

MP  (Fig.  53),  which  pro- 
duces greater  or  less  effect, 
according  to  the  position  of 
the  line  of  apsides  in  re- 
lation to  the  line  of  syzy- 
gies.  Let  FH  (Fig.  54) 
be  the  line  of  apsides  of  the  moon's  orbit  about  the  earth, 


VAKIATION.    '  105 

E ;  and  suppose  the  sun  to  be  in  the  direction  A.  Then  AC 
is  the  line  of  syzygies,  and  the  two  lines  coincide.  The 
moon's  gravity  toward  E  is  diminished  at  F  and  H,  as  it  always 
is  when  in  the  line  of  syzygies.  But  at  F,  it  is  diminished  less 
than  ever,  because  there  is  the  least  difference  of  distances,  AE 
and  AF ;  while  at  H,  it  is  diminished  more  than  ever,  because 
the  difference  of  distances  AE  and  AH  is  the  greatest  possible. 
Hence,  F  is  less  separated  from  E,  and  H  more  separated  from 
E,  than  in  any  other  situation.  The  same  would  be  true,  if  the 
sun  were  in  the  direction  of  C.  Therefore,  when  the  line  of 
apsides  coincides  with  the  line  of  syzygies,  the  moon's  orbit  is 
most  eccentric. 

Again,  suppose  the  sun  to  be  in  the  direction  B  or  D ;  in 
other  words,  that  the  line  of  apsides  is  in  quadrature.  Then, 
the  gravity  of  the  moon  toward  E  is  increased  at  F  and  H,  as 
it  always  is  when  in  quadrature.  But  at  F,  its  increase  is  the 
least  possible,  because  the  obliquity  of  FB  to  EB  is  the  least 
possible ;  while  at  H,  the  increase  is  the  greatest,  because  the 
obliquity  of  HB  to  EB  is  the  greatest.  Hence,  HE  is  less, 
compared  with  FE,  than  in  any  other  position.  Therefore,  the 
eccentricity  is  least  when  the  line  of  apsides  is  in  quadrature. 
The  greatest  evection  is  1°  20'. 

188.  Variation. — Another  correction  is  applied  on  account 
of  the  alternate  changes  of  velocity  caused  by  the  sun.  This 
change  of  velocity  is  called  variation.  It  is  produced  by  the 
tangential  disturbance  MO  (Fig.  53).  From  D  to  A,  it  con- 
spires with  the  motion  of  the  moon,  and  accelerates  it.  From 
A  to  B,  it  is  directed  backward,  and  retards  the  moon's  motion. 
From  B  to  0  it  accelerates,  and  from  C  to  D  it  retards.  It 
might  be  supposed  that  because  the  sun  attracts  toward  S,  this 
would  act  against  the  moon's  motion  in  going  from  B  to  C, 
and  thus  retard  it ;  and  with  it  from  C  to  D,  and  accelerate  it. 
But  the  disturbing  action  is  not  the  absolute,  but  the  relative 
attraction.  From  B  to  C,  the  sun  attracts  the  moon  less  than 
it  does  the  earth  ;  and  the  effect  is  the  same  as  if  it  exerted  no 
attraction  on  the  earth,  and  urged  the  moon  in  the  opposite 
direction — that  is,  toward  C.  Hence,  the  moon's  velocity  is 
alternately  accelerated  and  retarded  in  the  successive  quad- 


106      KETROGRADATION  OF  THE  MOON'S  NODES. 

rants,  causing  the  greatest  equation  about  35°  from  the  quad- 
ratures B  and  D.     The  variation  at  its  maximum  is  about  32'. 

189.  Annual  equation. — This  is  a  change  in  the  moon's 
motion,  arising  from  the  greater  and  less  distance  of  the  sun  at 
different  seasons  of  the  year.     The  disturbing  action  of  the  sun 
is  greatest  when  it  is  nearest — that  is,  at  perihelion;  and  it  is 
least  when  it  is  most  distant,  or  at  aphelion.     This  inequality 
is  called  the  annual  equation,  since  it  passes  through  all  its 
changes  in  a  year.     It  amounts  to  about  II7. 

1 9O.  /Smaller  equations. — The  foregoing  are  the  largest  in- 
equalities of  the  moon's  motion,  which  require  corrections  to 
be  made  for  finding  its  true  place.     There  is  a  large  number  of 
smaller  ones,  for  which  allowance  must  be  made,  in  order  to 
obtain  the  moon's  longitude  for  a  given  time,  with  the  utmost 
exactness.     By  the  most  complete  tables  of  the  moon  now  in 
use,  its  place  can  be  determined  within  3'' '. 

191.  Apsides  of  the  moon's  orbit. — The  line  of  apsides  ad- 
vances— that  is,  moves  forward — from  west  to  east.     This  is  a 
disturbance  produced  by  the  sun,  and  is  explained  in  the  same 
manner  as  the  advance  of  the  earth's  apsides  (Art.  148).     The 
attraction  of  a  body  external  to  the  orbit  always  tends  to  pro- 
duce this  effect.     Though  the  sun  makes  the  moon's  gravity  to 
the  earth  sometimes  greater,  and  sometimes  less,  yet  it,  on  the 
whole,  diminishes  it  (Art.  183).     Without  any  disturbing  in- 
fluence, the  moon  would  always  describe  the  same   elliptic 
orbit.     But  as  it  approaches  one  of  its  apsides,  it  is,  in  general, 
not  sufficiently  drawn  in  toward  the  center  to  cut  the  former 
line  of  apsides  at  right  angles ;  but  it  makes  right  angles  with 
a  radius  vector  a  little  further  on,  which,  therefore,  becomes 
the  new  line  of  apsides.     The  apsides  of  the  earth's  orbit  ad- 
vance with  exceeding  slowness  (Art.  147);   but  the  sun's  dis- 
turbing power  is  so  great,  that  those  of  the  moon's  orbit  shift 
their  place  more  than  3°  in  each  sidereal  month,  and,  therefore, 
make  a  complete  revolution  in  about  9  years. 

192.  Retrogradation  of  the  moon's  nodes. — In  Art.  183,  it 


EETROGRADATION  OF  THE  MOON'S  NODES. 


107 


\vas  noticed  that  the  sun's  oblique  action  on  tlie  moon  in  the 
plane  of  its  orbit  causes  increase  of  its  gravity  toward  the 
earth.  As  the  moon's  orbit  does  not  coincide  vvTith  the  ecliptic, 
the  sun  exerts  another  oblique  action — namely,  out  ofite  orbit, 
toward  the  ecliptic.  This  disturbance  causes  the  nodes  to 


retrograde. 


Fig.  55. 


Let  MN"  (Fig.  55)  represent  a  short  arc  of  the  ecliptic,  and 
AB  an  arc  of  the  moon's  path,  intersecting  ejach  other  in  N", 
the  descending  node.  When  the  moon  is  at  L,  moving  toward 
N,  the  sun,  at  a  great  distance,  attracts  the  moon  in  a  line 
slightly  oblique  to  M!N".  One  component  of  this  force  is  paral- 
lel to  iTN",  the  other  perpendicular  to  it.  Let  U  be  the  dis- 
tance through  which  the  latter  would  move  the  moon,  while  it 
would  pass  over  La  by  its  inertia  alone.  The  resultant  is  L<?, 
cutting  the  ecliptic  in  £T.  Again,  after  passing  the  node,  sup- 
pose the  sun's  action  would  move  the  moon  over  Id,  while  it 
would  describe  le  by  its  inertia.  Then,  by  their  joint  action, 
the  moon  will  describe  If,  which  produced  makes  the  node  at 
W.  Thus,  the  node  is  made  to  move  in  a  direction  contrary 
to  that  of  the  moon,  both  when  approaching  a  node  and  when 
departing  from  it. 

The  retrograde  motion  just  described  occurs  when  the  moon 
is  in  that  half  of  its  orbit  nearest  the  sun.  When  it  is  in  the 
opposite  half,  the  action  is  reversed,  because  the  moon,  being 
less  attracted  than  the  earth,  is,  as  it  were,  pushed  obliquely 
from  the  earth.  The  nodes  are,  therefore,  made  to  advance, 
instead  of  retrograding,  whenever  the  moon  passes  either  of 


108  ACCELERATION    OF    THE    MOOJST. 

them  on  the  side  of  its  orbit  furthest  from  the  sun.  But  this 
effect  is  less  than  the  other ;  so  that,  on  the  whole,  the  nodes 
have  a  retrograde  motion. 

The  nodes  of  the  moon's  orbit  retrograde  at  the  rate  of 
19°  35'  each  year,  thus  completing  a  revolution  in  18.6  years. 

193.  Disturbance  of  the  inclination  of  the  moon's  orbit. — 
When  the  moon  approaches  a  node,  the  inclination  of  its  orbit 
to  the  ecliptic  is  increased ;  for  LN'M  is  greater  than  the  in- 
terior angle  LNM.  And  the  inclination  is  diminished  as  the 
moon  leaves  a  node,  since  IW7$  is  less  than  the  exterior  angle 
IN'N.  These  alternate  changes  nearly  balance  each  other,  and 
leave  the  mean  value  of  the  inclination  almost  constant — 
namely,  5°  8'  55"  (Art.  153). 

1  94.  Periodical  and  secular  inequalities. — The  inequalities 
in  the  moon's  motion,  which  have  been  described,  pass  through 
all  their  changes  in  a  short  period,  as  a  month,  a  year,  or  a 
few  years  at  most.  These  are  called  periodical.  But  there  are 
others,  whose  periods  extend  through  many  centuries  or  ages. 
These  are  called  secular.  Some  minute  secular  disturbances  in 
the  solar  system  run  on  in  the  same  direction  for  an  indefinite 
number  of  centuries. 

195.  The  acceleration  of  the  moon's  mean  motion. — This  is 
an  interesting  example  of  secular  inequality.     The  period  of  a 
lunation  is  now  sensibly  shorter  than  it  was  before  the  Chris- 
tian era.     This  is  ascertained  by  comparing  the  recorded  date 
of  an  eclipse  which  occurred  in  721  before  Christ  with  the 
time  of  any  recent  eclipse.     The  whole  interval,  if  divided  by 
the  present  mean  length  of  a  lunation,  leaves  a  considerable 
remainder.     The  acceleration  amounts  to  about  10"  in  a  cen- 
tury. 

196.  Its  cause. — It  has  been  stated  that  the  sun  diminishes 
the  moon's  gravity  toward  the  earth  (Art.  183).     The  amount 
of  this  diminution  depends,  in  part,  on  the  eccentricity  of  the 
earth's  orbit.     From  the  time  of  the  earliest  observations,  the 
earth's  orbit  has  been  slowly  approaching  a  circle,  and  will 


ECLIPSE   MONTHS.  109 

continue  to  do  so  for  many  centuries  to  come.  So  long  as  the 
eccentricity  of  the  earth's  orbit  is  diminishing,  the  sun's  dis- 
turbing action  on  the  moon  diminishes  also.  The  moon,  there- 
fore, being  less  drawn  away  from  the  earth,  describes  a  smaller 
orbit,  and,  consequently,  in  a  shorter  time.  In  the  course  of 
ages,  the  earth's  orbit  will  reach  the  limit  of  its  change,  and 
begin  to  grow  more  eccentric.  The  moon's  orbit  will  then 
commence  to  enlarge,  and  will,  therefore,  require  a  longer 
time  to  be  described. 


CHAPTER  XII. 

•ECLIPSES  OF  THE  MOON. — ECLIPSES  OF  THE   SUN. 

197.  General  relations  in  eclipses. — The  moon  is  eclipsed, 
when  it  is  obscured  wholly  or  in  fepart  by  the  earth's  shadow. 
It  can  occur,  therefore,  only  at  opposition,  or  full  moon.     The 
sun  is  eclipsed,  when  it  is  either  wholly  or  partially  concealed 
from  view  by  the  moon  coming  between  it  and  the  earth.     This 
can  happen  only  at  conjunction,  or  new  moon. 

If  the  moon's  orbit  and  the  ecliptic  were  coincident  planes, 
there  must  be  an  eclipse  of  the  moon  at  every  full  moon,  and 
an  eclipse  of  the  sun  at  every  new  moon  ;  for  at  those  times 
the  three  bodies  would  be  in  a  straight  line.  But  as  the  moon's 
orbit  and  the  ecliptic  make  an  angle  of  5°  with  each  other,  the 
moon  generally  passes  opposition  and  conjunction  so  far  north 
or  south  of  the  sun,  that  there  is  no  eclipse.  That  an  eclipse 
may  occur,  the  syzygies  must  happen  near  the  line  of  nodes,  so 
that,  as  the  moon  comes  into  conjunction  or  opposition,  some 
parts  of  the  three  bodies  may  be  in  a  straight  line. 

198.  Eclipse  months. — As  there  are  two  nodes  on  opposite 
sides  of  the  heavens,  the  sun  in  its  annual  progress  must  pass 
through  both  of  them  every  year,  at  intervals  of  about  six 
months.     And  as  the  moon  comes  into  the  line  of  syzygiee 
every  two  weeks,  the  sun  will  certainly  be  near  enough  to  a 


110  EAKTH'S  SHADOW. 

node  for  one  or  two  eclipses,  and  possibly  for  three,  every  six 
months.  Thus,  the  eclipses  of  any  year  always  occur  in  clus- 
ters, at  opposite  seasons.  If  two  or  three  are  in  January,  the 
others  are  in  July.  These  are  called  the  node  months  of  that 
year.  In  1866,  for  example,  the  node  months  are  parts  of 
March  and  April,  and  parts  of  September  and  October.  On 
account  of  the  retrograde  motion  of  the  nodes,  the  sun  passes 
from  a  node  to  the  same  one  again  in  less  than  a  year,  so 
that  the  node  months  occur  earlier  each  successive  year  per- 
petually. 

\  99.  Eclipse  of  the  moon. — "When  the  moon  is  eclipsed, 
there  is  nothing  interposed  to  hide  it  from  our  view ;  but  it 
merely  falls  into  the  shadow  of  the  earth,  and  is  obscured. 
This  obscuration  may  possibly  continue  for  several  hours. 

2OO.  Form  and  angle  of  the  earth's  shadow. — As  the  sun 
is  larger  than  the  earth,  and  both  are  spheres,  the  tangents 
drawn  from  one  to  the  other,  along  the  corresponding  edges, 
will  converge  and  form  a  cone.  Thus  (Fig.  56),  let  AA'  be 
the  sun,  and  BB'  the  earth ;  then  BB'C  is  the  conical  shadow ; 
and  rays  of  light  from  AA',  moving  in  straight  lines,  can  not 
enter  any  part  of  it.  The  axis  of  the  shadow,  EC,  is  the  ex- 
tension of  the  line  joining  the  centers  of  the  sun  and  earth. 
Since  the  light  is  entirely  excluded  from  the  cone  BB'C,  it  is 
often  called  the  total  shadow. 

Fig.  56. 


Join  AE;  then  the. exterior  angle  AES  =  ACE  +  EAC; 
/.  ACE  ==  AES  --  EAC.  But  AES  is  the  sun's  apparent 
serai-diarneter,  and  EAC  is  the  sun's  horizontal  parallax. 


LUNAR  ECLIPTIC   LIMIT.  Ill 

Therefore,  the  semi-angle  of  the  earths  shadow  is  equal  to  the 
sun's  apparent  semi-diameter  diminished  hy  its  horizontal 
parallax.  Calling  the  sun's  semi-diameter  d,  and  its  horizontal 
parallax  jp,  the  semi-angle  of  the  shadow  is  8  —  p.  8  =  16'  1".5, 
and  p  =  8".6  ;  /.  8  —  p  =  15'  52".9,  the  mean  value  of  the 
semi-angle  of  the  shadow. 

20 1.  Length  of  the  earth's  shadow. — In  the  triangle  ECB, 
right-angled  at  B,  as  we  know  EB  and  ECB,  EC  is  found  by 
the  proportion,  sin  (6  —  p)  :  rad  : :  3956  :  856,275,  the  length 
of  the  earth's  shadow  in  miles. 

Since  the  moon  is  238,650  miles  from  the  earth,  the  length 
of  the  earth's  shadow  is  more  than  3-J-  times  the  distance  from 
the  earth  to  the  moon  ;  and  the  moon,  when  eclipsed,  passes 
through  the  broader  part  of  it. 

202.  Angular  hreadth  of  the  section  traversed  hy  the  moon. 
— Let  h'h  be  a  part   of  the  moon's  orbit  supposed  to  pass 
through  the  axis  of  the  shadow  at  M.     Then  Mm  is  the  semi- 
diameter  of  the  section,  and  ME??i  its  angular  semi-diameter, 
which  is  to  be  found.     The  exterior  angle  E/rcB  =  ECra  -f 
CEra  ;  .•.  CEm.  =  EmB  —  ECm.     But  EmB  is  the  horizontal 
parallax  of  the  moon,  and  ~ECm  the  semi-angle  of  the  shadow. 
Call  the  moon's  parallax  P,  then  the  angular  semi-diameter  of 
the  shadow  =  P  -  (d  —  p\  or  P  +  p  —  6. 

P  -  57'  5",  and  6  —  p  =  15'  52".9  ;  .-.  P  +  p  -  -  d  = 
41'  12".l,  the  mean  semi-diameter  of  the  section. 

Since  the  moon's  semi-diameter  is  15'  33",  the  breadth  of 
shadow  where  the  moon  crosses  it  is  2J  times  the  breadth  of 
the  moon. 

203.  Lunar  ecliptic  limit. — The  distance  of  the  center  of 
the  earth's  shadow  from  the  node,  when  the  moon  at  opposi- 
tion would  only  touch  the  shadow,  is  called  the  lunar  ecliptic 
limit.     Let  (JN  (Fig.  57)  be  an  arc  of  the  ecliptic,  MN  an  arc 
of  the  moon's  orbit,  N  the  node,  Ca  the  semi-diameter  of  the 
shadow,  and  «M  that  of  the  moon  when  it  only  touches  the 
shadow  at  opposition.     Then  CIST  is  the  ecliptic  limit.     In  the 
spherical  triangle  CMK,  right-angled  at  M,  1ST  being  known, 


112  DIMENSIONS   OF   PENUMBRA. 

and  also  Ca  and  «M,  we  have,  by  Napier's  rule,  rad  x  sin  CM 
=  sin  CN  x  sin  IS" ;  from  which  C~N  is  obtained.  Since  K. 
C«,  and  aM.  are  all  variable,  C~N  must  also  vary.  Its  greatest 
value  is  12°  24/,  beyond  which  an  eclipse  is  impossible.  Its 
least  value  is  9°  24'  within  which  an  eclipse  can  not  fail  to 
occur. 

Fig.  57. 


2O  4.  Magnitude  of  eclipse. — The  mere  contact  of  the  moon 
and  earth's  shadow  at  the  ecliptic  limit  is  called  an  appulse. 
If  the  moon  is  obscured  only  in  part,  the  phenomenon  is  called 
a  partial  eclipse.  It  is  a  total  eclipse  when  the  moon  is  en- 
tirely enveloped  in  the  shadow.  If  its  center  passes  through 
the  axis  of  the  shadow,  there  is  a  central  eclipse. 

2O  5.  TJie  earth's  penumbra. — If  tangents  be  drawn  across 
the  opposite  sides  of  the  sun  and  earth,  as  AA',  A'A  (Fig.  56), 
they  diverge,  and  inclose  a  space  around  the  total  shadow, 
called  the  penumbra,  or  partial  shadow.  Its  form  is  the  frus- 
tum of  a  cone,  and  it  extends  to  an  infinite  distance  beyond 
the  earth.  Within  the  penumbra,  and  outside  of  the  shadow, 
there  is  light  from  a  part  of  the  sun  only,  while  the  other  part 
is  concealed  by  the  earth.  Thus,  at  a  point  between  BC  and 
BA  produced,  it  is  obvious  that  the  limb  of  the  sun  near  A' 
could  not  shine,  because  the  light  would  be  intercepted  by  the 
opposite  side  of  the  earth  near  B.  The  vertex  of  the  penumbra 
is  between  the  earth  and  sun,  at  C'. 

2O6.  Dimensions  of  the  penumbra. — The  semi-angle  of  the 
penumbra  is  A'C'C  (Fig.  56),  which  is  equal  to  AC'S.  And  the 
external  angle  AC'S  -  EAC'  +  C'EA.  But  EAC'  is  the  sun's 
horizontal  parallax  =  p  ;  and  C'EA  is  the  sun's  apparent  semi- 


SOLAR   AND    LUNAR   TABLES.  113 

diameter  =  (5;   therefore,  the   semi-angle  of  the  penumbra  = 
p  +  cJ. 

The  semi-diameter  of  the  section  of  the  penumbra  through 
which  the  moon  passes  is  AM,  and  its  angular  or  apparent 
semi-diameter  is  //EM.  And  AEM,  being  external  to  the  tri- 
angle AEO',  equals  EC'A  +  EAC'.  But  EAC'  is  the  moon's 
horizontal  parallax  =  P ;  and  EC'A  —  p  +  $ ;  therefore,  the 
apparent  semi-diameter  of  the  earth's  penumbra  —  P  -f  p  +  <V 
At  mean  values,  this  equals  1°  13'  19",  which  is  nearly  5  times 
the  semi-diameter  of  the  moon. 

207.  Effect  of  the  penumbra. — On  account  of  the  penum- 
bra, the  edge  of  the  total  shadow  is  not  sharply  denned,  but 
shades  off  into  the  full  light  by  slow  degrees,  so  that  the  moon 
passes  over  rather  more  than  its  own  breadth  after  entering  the 
penumbra,  before  it  reaches  the  total  shadow.     This  circum- 
stance renders  the  exact  moment  of  beginning  or  end  of  a  lunar 
eclipse  uncertain. 

208.  Effects  of  the  earth's  atmosphere. — It  is  found,  by  cal- 
culation, that  the  sun's  light  which  traverses  the  lowest  parts 
of  the  earth's  atmosphere  would  be  so  much  refracted  as  to 
meet  the  axis  of  the  shadow  before  reaching  the  moon.     Hence, 
the  whole  disk  of  the  moon  is  visible,  even  in  a  central  eclipse, 
and  appears  of  a  dull  red  color. 

Another  effect  is  the  enlargement  of  the  shadow.  The  light, 
which  passes  the  earth  near  its  surface,  and  would  immediately 
surround  the  shadow  if  there  were  no  atmosphere,  is,  in  part, 
obstructed,  and  in  part  diffused  through  the  whole  breadth  of 
the  shadow,  as  just  stated.  Therefore,  the  boundary  of  the 
shadow  is  enlarged.  To  make  its  computed  diameter  agree 
best  with  the  observed  diameter,  it  is  necessary  to  add  ^V 

209.  Solar  and  lunar  talles. — In  order  to  determine  the 
circumstances  of  any  particular  eclipse,  tables  are  needed  which 
will  give  for  that  time  the  sun's  and  moon's  hourly  motions, 
their  parallaxes,  and  their  apparent  semi-diameters.    Such  tables, 
of  the  most  accurate  kind,  are  published  in  the  Nautical  Alma- 
nac for  each  year,  and  several  years  in  advance. 

8 


LUNAR  ECLIPSE. 

2 1 0.  The  moon's  relative  orbit. — The  center  of  the  earth's 
shadow  moves  in  the  ecliptic  at  the  same  rate  as  the  sun,  about 
1°  per  day;  while  the  moon  moves  in  its  orbit  about  13°  per 
day.  To  reduce  these  two  motions  to  one,  the  relative  orbit  is 
substituted  for  the  real  one,  in  the  following  manner.  Let  NG 
(Fig.  58)  be  an  arc  of-  the  ecliptic,  ~Ng  an  arc  of  the  moon's 
orbit,  N  the  ascending  node,  A  the  place  of  the  shadow's  cen- 
ter, and  a  that  of  the  moon's  center,  at  the  time  of  opposition. 
While  A,  in  one  hour,  moves  to  A7,  suppose  a  to  move  to  g. 
Then  A!g  represents  the  distance  and  relative  direction  of  the 
centers  at  the  end  of  an  hour  after  opposition.  If  gd  be  drawn 
equal  and  parallel  to  A7 A,  then  Ad  has  the  same  length  and 
direction  as  A!g.  We  may,  therefore,  suppose  A,  the  center  of 
the  shadow,  to  have  remained  at  rest,  and  a,  the  moon's  center, 
to  have  moved  to  d  in  one  hour ;  in  which  case,  Yad  would  be 
the  relative  orbit,  id  (=  hg)  is  the  moon's  hourly  motion  in  lati- 
tude, and  ai  (=  ah  —  AA')  is  the  difference  of  hourly  motions 
in  longitude. 

Fig.  58. 


The  inclination  of  the  relative  orbit  to  the  ecliptic  is  found 
by  the  right-angled  triangle  dai,  in  which  ai  and  di  being 
known,  the  angle  dai,  or  its  equal,  6?FD,  is  computed. 

The  change  from  the  true  to  the  relative  orbit  is  greatly  ex- 
aggerated in  the  figure.  If  truly  represented,  ag  would  be  13 
times  as  long  as  AA'. 

211.  Times  of  beginning,  middle,  and  end  of  a  lunar 
eclipse,  ~by  projection. — Let  ND  (Fig.  59)  represent  an  arc  of  the 
ecliptic,  and  A  the  center  of  the  shadow  at  opposition.  With 
any  convenient  scale  of  equal  parts,  lay  off  from  A  the  minutes 
of  hourly  motion  of  the  moon  from  the  sun — namely,  AB,  BC, 
AD,  etc.,  and  divide  them  into  as  small  fractions  of  an  hour  as 
is  desired.  Then  draw  a  circle  with  the  radius  Ao,  equal  to 


MIDDLE   AND   OPPOSITION. 


115 


the  minutes  in  the  semi-diameter  of  the  shadow.  Lay  off  A.a 
perpendicular  to  CD,  equal  to  the  moon's  latitude  at  opposi- 
tion. Then  a  is  the  moon's  center  at  that  time.  Through  a 
draw  Ny,  making  N  equal  to  the  inclination  of  the  relative 
orbit.  Draw  Ara0  perpendicular  to  ~Nf.  At  the  middle  of 
the  eclipse,  the  moon's  center  is  at  m,  because  Km  bisects  the 
chord  of  the  circle.  From  m  draw  mM.  perpendicular  to  OT). 
The  parts  of  hourly  motion  between  M  and  A  show  how  long 
before  opposition  the  middle  of  the  eclipse  occurs. 


Fig.  59. 


Take  a  line  equal  to  the  sum  of  the  semi-diameters  of  the 
shadow  and  the  moon,  place  one  end  at  A,  and  mark  the  points 
c  and/",  with  the  other  end  on  the  moon's  path.  With  a  radius 
equal  to  the  semi-diameter  of  the  moon,  draw  the  circles  around 
c  and/",  which  will  touch  the  shadow.  The  eclipse  begins 
when  the  moon's  center  is  at  c,  and  ends  when  at  f.  Next 
draw  the  perpendiculars  cF,  /G-,  and  we  have  on  the  scale  of 
time  the  interval  FA  between  the  beginning  and  opposition, 
and  AG-  between  opposition  and  the  end. 

Finally,  if  the  latitude  is  so  small  that  the  moon  falls  entirely 
into  the  shadow,  making  Ac',  A/*',  each  equal  to  the  difference 
of  the  two  semi-diameters,  mark  the  points  cr  and/*'  as  before. 
Then  the  perpendiculars,  c'H  and/*' K,  mark  the  times  of  the 
beginning  and  end  of  the  total  eclipse. 

212.  The  middle  of  the  eclipse,  how  related  to  the  opposi- 
tion.— In  the  projection  just  described,  IS"  is  the  ascending 


116  ECLIPSE   OF   THE    SUN. 

node,  and  the  moon  passes  the  node,  X,  before  it  reaches  the 
opposition,  a  ;  in  which  case,  the  middle  of  the  eclipse  at  m 
precedes  the  opposition  at  a.  This  is  true  at  either  node  ;  the 
middle  of  the  eclipse  precedes  opposition,  if  the  passage  of  the 
node  precedes  it ;  but  the  middle  is  later  than  opposition,  if  the 
passage  of  the  node  is  later. 

213.  Times  of  beginning,  middle,  and  end  of  a  lunar 
eclipse  by  computation. — The  same  results  may  be  obtained 
with  greater  accuracy  by  trigonometry. 

As  Aa  and  Am  are  perpendicular  respectively  to  ND  and 
N/1,  the  angle  a  Am  is  equal  to  AN#,  the  angle  of  the  relative 
orbit.  The  moon's  latitude,  Aa,  being  known,  and  the  angle 
«Aw,  compute  Am  ;  then  by  Am  and  AmM  (—  aAm)  find  AM, 
and  change  it  into  time  by  the  proportion,  hourly  motion  in 
longitude  of  moon  from  shadow  :  MA  : :  1  hour  :  time  of  pass- 
ing over  MA.  Thus,  the  time  of  the  middle  of  the  eclipse  is 
obtained. 

Am  and  Ac  being  known,  the  angle  mAc  is  calculated; 
which  subtracted  from  mAM.  (complement  of  aAm)  leaves 
cAN.  Hence,  in  the  triangle  A<?F,  A c  and  the  angle  cAF  fur- 
nish FA,  which,  changed  to  time  as  before,  determines  the  time 
when  the  eclipse  begins.  In  the  same  manner,  by  the  triangle 
Ac'H,  the  time  of  the  beginning  of  the  total  eclipse  is  found. 
No  additional  calculation  is  necessary  for  the  end ;  for  the  in- 
terval between  the  beginning  and  middle  is  equal  to  that  be- 
tween the  middle  and  the  end. 

214.  Digits  eclipsed. — The  v  magnitude   of   an  eclipse  is 
usually  expressed  in  digits,  or  12ths  of  the  moon's  diameter. 
The  distance  from  w,  the  inner  edge  of  the  moon,  to  <?,  the  edge 
of  the  shadow,  is  divided  into  parts,  each  equal  to  -^  of  nl. 
The  number  of  such  parts  contained  in  no  expresses  the  digits 
eclipsed.     If  the  digits  eclipsed  equal  or  exceed  12,  the  eclipse 
is  total. 

215.  Eclipse  of  the  sun. — An  eclipse  of  the  sun  is  of  a  dif- 
ferent character  from  an  eclipse  of  the  moon.     When  the  moon 
is  eclipsed,  it  is  obscured  by  the  earth's  shadow  falling  on  it. 


LENGTH   OF   MOON^S  SHADOW.  117 

The  moon  itself  is  affected.  But  the  sun  is  said  to  be  eclipsed 
when  the  moon  intervenes  between  it  and  the  earth,  and  hides 
it  from  our  view.  The  sun  itself  suffers  no  change,  but  we  are 
placed  in  circumstances  which  prevent  our  seeing  it.  The 
phenomenon  would  more  properly  be  called  an  oocultation  of 
the  sun. 

216.  Form  and  angle  of  the  moorfs  shadow. — The  moon's 
shadow,  like  the  earth's,  is  a  cone,  surrounded  by  a  penumbra 
of  infinite  extent.  Let  AR  (Fig.  60)  be  the  sun,  BC  the  moon, 
and  K,  the  vertex  of  its  conical  shadow.  The  exterior  angle 
SDR  =-  DRK  +  DKR;  >  DKR  -  SDR  -  -  DRK.  Now, 
SDR  is  readily  found,  being  the  apparent  semi- diameter  of  the 
sun  as  seen  from  the  moon.  It  is  larger  than  as  seen  from  the 
earth,  in  the  inverse  ratio  of  distances,  or  as  400  :  399,  nearly. 
The  angle  DRK  is  the  sun's  horizontal  parallax  at  the  moon. 
On  account  of  distance,  it  is  larger  than  at  the  earth,  nearly  in 
the  ratio  of  400  :  399 ;  but,  on  account  of  the  moon's  size,  it  is 
less  in  the  ratio  of  their  diameters,  2160  :  Y912.  The  sun's 
horizontal  parallax  at  the  earth,  when  thus  modified,  gives  the 
angle  DRK.  Therefore,  DKR,  the  semi-angle  of  the  moon's 
shadow,  is  found.  Its  mean  value  is  16'  1".6,  about  the  tame 
as  the  sun's  apparent  semi-diameter. 

Fig.  60. 


217.  Length  of  the  moorfs  shadow. — In  the  triangle  DKC, 
right-angled  at  C, 

sin  DKC  :  rad  : :  DC  :  DK, 

the  length  of  the  moon's  shadow.     Its  mean  length  is  231,690 
miles,  not  quite  sufficient  to  reach  to  the  earth's  surface. 

When  the  moon  is  nearest  to  the  earth,  and  the  earth  at  the 


118  SOLAK  ECLIPTIC  LIMIT. 

same  time  is  furthest  from  the  sun,  the  shadow  is  long  enough 
to  reach  about  14,500  miles  beyond  the  earth's  center. 

218.  Greatest  breadth  of  section  on  the  earth. — In  the  case 
just  mentioned,  if  the  shadow  is  directed  toward  the  earth's 
center,  its  section  at  the  surface  is  the  greatest  possible.     To 
find  its  diameter  en,  compute  the  angle  eld, 

eT  :  TK  : :  sin  eKT  :  sin  TeK, 
and  eTd  =  eKT  +  TeK.     Then,  as 

360°  :  eld  : :  earth's  circumference  :  ed. 

This,  when  greatest,  is  about  85  miles,  and  therefore  the 
diameter  of  the  section  is  170  miles.  Within  this  circle  there 
is  witnessed  a  total  eclipse  of  the  sun. 

219.  The  moon's  penumbra,  and  its  greatest  section  on  the 
earth. — The  crossing  tangents,  ACH,  RBG,  etc.,  include  the 
penumbra.     Its  semi-angle  is  BID,  which  is  equal  to  IRD  4- 
IDR.     But  IKD  is  the  sun's  horizontal  parallax  at  the  moon, 
and  IDR  is  the  sun's  apparent  semi-diameter  at  the  moon. 
Therefore,  BID  is  known.     To  this  add  IGD,  the  moon's  hori- 
zontal parallax,  and  the  sum  equals  GDT.     Hence,  in  the  tri- 
angle GDT  we  have  GT,  TD,  and  the  angle  GDT,  by  which 
GTD  is  computed.     From  this,  GH,  the  diameter  of  the  pe- 
numbra on  the  earth,  is  obtained,  as  in  the  preceding  article. 
Its  greatest  diameter  is  4,500  miles. 

2  2O.  Solar  ecliptic  limit. — The  distance  of  the  sun's  center 
from  the  node,  when  the  moon's  penumbra  at  conjunction 
would  only  touch  the  earth  in  passing,  is  called  the  solar  eclip- 
tic limit.  It  is  obtained  by  first  finding  the  distance  between 
the  sun's  and  moon's  centers  at  the  given  time.  Let  S  (Fig. 
61)  be  the  sun's  center,  E  the  earth's,  and  M  the  moon's.  It  is 
obvious  that  the  limit  occurs  when  the  moon's  disk  just  touches 
AB,  the  extreme  solar  ray  that  meets  the  earth.  The  angular 
distance  between  the  centers  of  the  sun  and  moon  at  that  time 
is  the  angle  SEM.  But  SEM  -  SEA  +  AEC  +  CEM.  SEA 
is  the  sun's  semi-diameter  =  6.  CEM  is  the  moon's  semi-diame- 
ter =  d.  The  angle  AEC  (in  the  triangle  EAC)  =  ECB  -  CAE. 


MAGNITUDE   OF   ECLIPSE,  119 

But  ECB  is  the  moon's  horizontal  parallax  =  P ;  and  CAE  is 
the  sun's  horizontal  parallax  =  p.  Therefore,  the  distance  be- 
tween the  centers,  SEM  —  (?  -f  <#  +  P  — p  •  that  is,  the  sum 
of  the  semi-diameters  of  the  sun  and  moon,  addecLjo  the  differ- 
ence of  their  parallaxes. 

Fig.  61. 


V 


Representing  this  distance  by  CM  (Fig.  57),CN  is  computed 
as  in  Art.  203.  At  the  maximum,  it  is  found  to  be  18°  36' ; 
and  beyond  that,  an  eclipse  is  impossible.  Its  minimum  value 
is  15°  20' ;  and  within  that,  there  cannot  fail  to  be  an  eclipse. 

221.  Magnitude  of  eclipse. — If  the  eye  of  the  observer  were 
at  the  vertex  of  the  total  shadow  of  the  moon,  it  is  plain  that 
the  moon's  disk  would  exactly  cover  the  sun's.  And  as  the 
moon  appears  to  our  unaided  vision  to  be  of  the  same  size  as 
the  sun,  this,  of  itself,  shows  that  the  cone  of  the  shadow  has  a 
length  sufficient  to  reach  about  to  the  earth,  as  proved  (Art. 
217).  But  the  moon's  semi- diameter  is  sometimes  greater  than 
the  sun's,  and  sometimes  less.  When  greater,  the  eclipse  is 
total  to  all  those  places  which  fall  within  the  section  of  the 
shadow  as  it  crosses  the  earth.  When  less,  the  eclipse  is  annu- 
lar to  places  lying  sufficiently  near  the  path  of  the  axis  of  the 
shadow.  It  is  called  annular,  because  a  ring  of  the  sun's  disk 
is  seen  about  the  moon  (Fig.  62).  An  eclipse,  whether  \otal  or 
annular,  is  central  at  all  places  where  the  axis  of  the  shadow 
tails,  or  to  which  it  points.  If  only  the  penumbra  passes  a 
place,  the  eclipse  there  impartial.  The  annular  eclipse  belongs 
to  the  class  of  partial  eclipses. 

If  the  total  shadow  reaches  the  earth  at  all,  yet  its  section  is 
small,  compared  with  that  of  the  penumbra  (Arts.  218  and 


120  VELOCITY   OF   THE    SHADOW. 

210).  Hence,  at  a  given  place,  while  partial  solar  eclipses 
occur  frequently,  probably  one  or  two  every  year,  a  total 
eclipse  is  extremely  rare,  perhaps  not  one  in  a  century. 

Fig.  62. 


It  is  possible  for  an  eclipse  to  be  annular  to  those  places 
where  it  is  seen  in  the  morning  or  evening,  and  total  to  those 
in  which  it  is  seen  near  noon ;  for  on  the  meridian,  the  moon 
appears  about  -^  larger  than  at  the  horizon  (Art.  165),  and 
might  cover  the  sun  in  one  case,  when  it  would  not  in  the 
other.  If  an  eclipse  thus  changes  its  magnitude  from  annular 
to  total,  and  then  to  annular  again,  while  crossing  the  earth,  it 
results  from  the  fact  that  the  moon's  shadow  is  too  long  to 
reach  the  nearest  part  of  the  earth's  surface,  and  not  long 
enough  to  reach  its  center. 

222.  Velocity  of  the  shadow. — The  hourly  motion  of  the 
moon  from  the  sun  is  about  30'.  This  arc  equals  2,080  miles 
of  absolute  motion  of  the  moon  in  its  orbit.  The  shadow  may 
be  considered  as  having  the  same  velocity  as  the  moon.  There- 
fore, the  absolute  velocity  of  the  moon's  shadow  on  the  earth  is 
2,08' >  miles  per  hour,  which  is  sufficient  to  carry  it  across  the 
earth's  disk  in  a  little  less  than  4  hours.  Relatively  to  the  sur- 
face, the  velocity  is  much  less  than  this.  At  the  equator,  the 


SOLAR   AND   LUNAR  ECLIPSES.  121 

velocity  of  surface  is  about  1,040  miles  per  hour,  one-half  that 
of  the  shadow,  and  both  motions  are  from  west  to  east.  Hence, 
at  the  equator,  the  shadow  passes  a  place  at  the  rate  of  about 
1,040  miles  per  hour,  when  it  falls  perpendicularly.  When  in- 
clined, as  at  morning  and  evening,  it  passes  more  swiftly,  in 
the  proportion  of  radius  to  the  sine  of  obliquity.  The  relative 
motion  is  also  greater  as  the  latitude  increases,  OTI  account  of 
the  slower  motion  of  the  surface.  When  an  eclipse  falls  within 
a  polar  circle,  the  shadow  and  the  observer  may  possibly  move 
in  opposite  directions,  so  that  the  relative  motion  would  be  the 
sum,  instead  of  the  difference,  of  the  real  motions. 

223.  Duration  of  total  and  annular  eclipses. — The  sun 
and  moon  differ  so  little  in  apparent  size,  and  the  velocity  of 
the  shadow  is  so  great,  that  the  duration  of  total  and  annular 
eclipses  is  necessarily  short.     It  is  seen  by  the  preceding  article 
that  the  rotation  of  the  earth  generally  reduces  the  relative  ve- 
locity ;  it  therefore  increases  the  duration.     The  greatest  con- 
tinuance of  a  total  eclipse  of  the  sun  is  about  8  minutes.     An 
annular  eclipse  may  continue  more  than  12  minutes. 

224.  Number  of  solar  and  lunar  eclipses. — If  an  eclipse  of 
the  sun  occurs  in  passing  each  node  in  a  certain  year,  the  lunar 
ecliptic  limit  is  so  small,  that  the  moon  may  escape,  an  eclipse 
at  both  the  previous  and  the  subsequent  oppositions.     In  this 
case,  there  would  be  but  two  eclipses  in  a  year,  both  solar. 
This  is  the  least  number. 

If,  however,  a  lunar  eclipse  occurs  very  near  a  node,  the 
solar  limit  is  so  large,  that  there  must  be  one,  and  there  may 
be  two  solar  eclipses  at  the  preceding  and  following  conjunc- 
tions. Thus,  there  may  be  as  many  as  six  eclipses  while  the 
sun  passes  the  two  nodes.  Another  one  may  possibly  occur 
before  twelve  months  have  elapsed,  in  consequence  of  the  back- 
ward motion  of  the  nodes.  Thus,  the  greatest  number  in  a 
year  is  seven,  of  which  five  are  of  the  sun,  and  two  of  the 
moon. 

225.  Relative  number  of  solar  and  lunar  eclipses. — Solar 
eclipses  are  more  numerous  than  lunar,  in  the  proportion  of 


122  THE   SAROS. 

their  ecliptic  limits — that  is,  nearly  as  3  :  2.  But,  because  one 
is  really  an  eclipse,  and  the  other  an  occupation,  eclipses  of 
the  moon  at  a  given  place  are  more  frequent  than  those  of 
the  sun.  An  eclipse  of  the  moon  is  visible  to  all  on  the  hem- 
isphere nearest  to  it,  without  regard  to  locality.  But  an 
eclipse  of  the  sun  is  not  seen  at  a  place,  unless  the  moon's 
shadow  falls  at  that  place. 

226.  Solar  and  lunar  eclipses  begin  on  opposite  sides. — As 
the  moon  moves  toward  the  east  much  faster  than  the  sun  or 
the  earth's  shadow,  we  determine  on  which  side  of  the  body 
a  solar  or  a  lunar  eclipse  begins,  by  simply  considering   the 
motion  of  the  moon.     In  a  lunar  eclipse,  the  moon  overtakes 
the  shadow  of  the  earth,  and,  of  course,  its  eastern  limb  enters 
the  shadow  first.     Hence,  a  lunar  eclipse   always    begins   on 
the  east  side  of  the  moon,  and  ends  on  the  west  side.     Bat 
in  a  solar  eclipse,  the  moon,  in  its  eastward  motion,  overtakes 
the  sun,  and  conceals  its  western  limb  first ;   so  that  a  solar 
eclipse  begins  on  the  west  side  of  the  sun,  and  ends  on   the 
east  side. 

227.  The  Saros. — This  name  is  given  to  the  cycle  of  13 
years  and  10  days,  within  which   there   is    a   return   of  the 
eclipses  of  preceding  cycles,  in  the  same  order,  and  of  nearly 
the  same  magnitude.    The  reason  for  this  return  of  eclipses  is, 
that  the  sun,  moon,  and  node,  return  to  very  nearly  the  same 
relations  to  each  other  in  the  period  just  named. 

The  return  of  the  moon  to  the  sun  (a  lunation)  occurs  223 
times,  and  the  return  of  the  sun  to  the  node  (a  synodical  rev- 
olution of  the  node)  occurs  19  times,  in  this  period  of  18  years 
and  10  or  11  days,  the  two  periods  differing  less  than  12 
hours  from  each  other.  As  the  sun,  moon,  and  node,  do  not 
resume  their  exact  relation  to  each  other,  the  series  of  eclipses 
in  one  cycle  will  vary  a  little  from  those  of  the  preceding ; 
and,  therefore,  after  a  number  of  cycles,  their  magnitude  will 
become  essentially  changed,  and  at  length,  one  after  another, 
they  will  disappear  from  the  cycle  entirely. 

This  period  was  used  by  the  Chaldeans  for  predicting  the 
returns  of  eclipses,  and  by  them  called  the  Saros. 


PHENOMENA   OF   A   SOLA'S    ECLIPSE.  123 

228.  Phenomena  of  a  total  eclipse  of  the  sun. — 

1.  The  corona. — This  is  a  luminous  halo  surrounding  the 

o 

moon  when  the  sun  is  entirely  hidden,  and  sometimes  presents 
a  radiated  appearance,  and  extends  from  the  moon's  edge  out- 
ward a  distance  equal  to  one-third  of  its  diameter,  fading 
gradually  to  the  shade  of  the  sky.  It  is  concentric  with  the 
sun,  rather  than  with  the  moon,  and  is  thought  to  indicate 
an  extensive  solar  atmosphere. 

2.  Rally's  bead*. — At  the  instant  when  the  fine  thread  of 
the  sun's  edge  is  just    appearing  or  disappearing,  it  is  often 
divided  up  into  a  series  of  separate  bright  points.     Being  first 
noticed  by  Sir  Francis  Baily,  they  are  known  as  Baily's  beads. 
The    appearance   is   by  some  attributed  to  the   light    of  the 
sun's  edge  coming  through  between    the   mountain   summits 
of  the  rough  outline  of  the  moon's  disk.     That  they  are  not 
always  seen,  may  arise   from  the  fact  that  the  limb  in  con- 
tact may,  in  some  cases,  be  much  less  serrated  by  mountains 
than  in  others. 

3.  jF7  lame-colored  protuberances. — Another  phenomenon,  very 
variable  in  its  aspect,  consists  of  irregular  projections,  which 
appear  here  and  there  around  the  disk  of  the  sun,  after  it  is 
wholly  in  occupation.   They  are  sometimes  broad,  and  of  small 
elevation ;  at  others,  they  extend  out  nearly  a  tenth  of  the  di- 
ameter of  the  sun — that  is,  to  the  height  of  80,000  miles,  and 
are  often  bent  at  a  considerable  angle.     Occasionally,  they  are 
entirely  detached  from  the  disk.     They  are  not  sufficiently  lu- 
minous to  be  seen  at  all  when  any  part  of  the  sun  is  visible. 
They  are   described   by  some   as    rose-colored,    by  others,    as 
flame-colored.     In  different   eclipses,  they  differ   entirely  in 
their  arrangement  and  general  aspect. 

The  most  probable  explanation  is,  that  they  are  cloud-like 
substances,  floating  above  the  visible  surface  of  the  sun. 

A  total  eclipse  of  the  sun  is  one  of  the  most  sublime  and  im- 
pressive phenomena  of  nature.  The  darkness  is  such,  that  the 
larger  planets  and  stars  appear  ;  and  yet  it  is  surprisingly  sud- 
den in  coming  and  going  ;  for  within  a  few  seconds  before  and 
after  the  total  darkness,  the  light  is  equal  to  that  of  hundreds 
of  full  moons.  A  chill  is  felt  like  that  of  night.  It  is  not 
strange  that  people  of  barbarous  countries  are  filled  with  con- 


124:  CALCULATION  OF   ECLIPSES. 

sternation  and  fear  by  the  occurrence  of  a  total  eclipse  of  the 
sun. 

229.  Eclipses  at  the  moon. — "When   we   witness   a  solar 
eclipse,  a  spectator  at  the  moon  would  notice  only  a  small, 
dimly-defined  circular  shadow  passing  over  the  earth's  disk. 
It  would  be  a  partial  eclipse  of  the  earth. 

But  when  we  see  a  total  lunar  eclipse,  the  phenomenon  at 
the  moon  would  be  one  of  great  interest,  and  of  very  strange 
appearance.  A  dim  red  light  from  all  parts  of  the  sun's  disk 
is  spread  over  the  moon,  being  refracted  thither  by  the  earth's 
atmosphere  (Art.  20S).  Hence,  a  spectator  there  would  see  the 
sun  expanded  out  into  a  thin  dull  red  ring,  surrounding  the 
earth,  and,  therefore,  having  nearly  four  times  the  usual  diam- 
eter of  the  sun's  disk. 

230.  True  form  of  shadows. — It  is  impossible,  in  ordinary 
diagrams,  to  present  the  shadows  of  the  earth  and   moon  in 
their  true  proportions.     The  distance  of  the  sun   is  so  very 
great,  compared  with  its  diameter,  that  the  shadows  are  ex- 
ceedingly slender,  having  a  length  about  110  times  the  diame- 
ter of  the  base.     Fig.  63  is  intended  to  exhibit  them  in  their 
true  forms.     A  is  the  earth,  and  B  the  moon,  having  just 
emerged  from  an  eclipse.     Only  one-half  of  the  whole  length 
of  the  shadow  of  each  is  presented.     Again,  on  another  scale, 
C  is  the  moon,  and  D  the  earth,  on  which  its  shadow  is  falling 
in  a  solar  eclipse. 

Fig.  63. 


231.  Calculation  of  eclipses. — Particular  instructions  are 
given  in  various  works  on  practical  astronomy  for  calculating 
all  the  circumstances  of  a  solar  or  a  lunar  eclipse.  Such  in- 
structions, with  examples  for  illustration,  may  be  found  in 
Mason's  Supplement  to  this  work. 


LONGITUDE   BY   THE   CHRONOMETER.  125 


CHAPTEE  XIII. 

METHODS  OF  DETERMINING  TERRESTRIAL   LONGITUDE. 

232.  Local  time. — Time  is  reckoned  at  every  place  from 
the  moment  when  the  sun  crosses  the  meridian  at  either  the 
upper  or  the  lower  culmination.     This  is  called  local  time  ;  for 
at  the  same  absolute  instant,  the  time  thus  reckoned  at  any 
place  diifers  from  that  on  every  other  meridian. 

233.  Connection  between  longitude  and  local  time. — The 
earth  turns  uniformly  on  its  axis  toward  the  east  through  15° 
every  hour.     Therefore,  a  place  lying  eastward  of  another  will 
have  the  sun  earlier  on  its  meridian,  and  consequently,  in 
respect  to  the  hour  of  the  day,  will  be  in  advance  of  the  other 
at  the  rate  of  one  hour  for  every  15°.     Thus,  to  a  place  15° 
east  of  Greenwich  observatory,  it  is  1  o'clock  P.  M.  when  it  is 
noon  at  Greenwich;  and  to  a  place  15°  west  of  that  meridian, 
it  is  11  o'clock  A.  M.  at  the  same  instant.     Hence,  the  differ- 
ence of  local  time  at  any  two  places  indicates  their  difference 
of  longitude. 

234.  Longitude  by  the  chronometer. — If  a  person  leaves 
London  with  a  chronometer  accurately  adjusted  to  Greenwich 
time,  and  travels  eastward  till  he  finds  his  own  time  slower 
than  the  local  time  of  the  place  by  Ih.  30m.,  then  he  knows 
the  place  to  be  22°  30'  E.  longitude.     For  15°  x  1±  =  22i°. 
On  the  contrary,  if  he  travels  westward,  and  at  length  finds 
his  time-piece  at  6h.  44m.,  when  the  local  time  is  4h.  32m. — in 
other  words,  that  his  Greenwich  time  is  2h.  12m.  too  fast — then 
the  longitude  of  the  place  is  33°  W.     In  the  same  manner, 
the  longitude  of  any  two  places  may  be  compared  with  each 
other. 

For  the  use  of  navigators,  chronometers  are  made  which  run 
with  very  great  accuracy,  and  may  be  relied  on  during  long 


126  LONGITUDE   BY   A   SOLAR   ECLIPSE. 

voyages.  There  is  always  a  probability,  however,  that  a  chro- 
nometer may  change  its  rate  somewhat,  when  it  comes  to  be 
transported  from  place  to  place.  It  is  therefore  safer  on  long 
voyages  to  use  several  chronometers,  and  employ  the  mean  of 
all  their  indications. 

235.  Longitude  ~by  a  lunar  eclipse. — In  one  respect,  a  lunar 
eclipse  is  very  favorable  for  the  comparison  of  longitudes.     It 
is  a  distant  phenomenon,  seen  at  the  same  absolute  instant  by 
all.     Hence,  any  difference  of  time  in  the  observations  at  dif- 
ferent places  is  entirely  due  to  difference  of  longitude. 

But  in  another  respect,  it  is  quite  unfitted  for  the  purpose. 
On  account  of  the  penumbra,  there  is  no  definite  edge  to  the 
shadow  which  passes  over  the  moon's  disk,  and  consequently 
there  is  great  uncertainty  as  to  the  time  of  beginning  or  end  of 
the  eclipse.  This  method  is  but  little  depended  on  for  accurate 
results. 

236.  Longitude  by  a  solar  eclipse. — In  both  the  above  par- 
ticulars, a  solar  eclipse  differs  from  a  lunar.     It  is  not  an  event 
at  a  distance,  seen  at  once  by  all,  but  on  the  earth's  surface, 
happening  to  one  place  at  one  instant,  and  to  another  place  at 
another.     The  time  of  beginning  or  end  of  a  solar  eclipse  de- 
pends on  the  position  of  the  observer. 

On  the  other  hand,  the  phenomenon  is  very  definite,  and 
the  moments  of  immersion  and  emersion  of  the  sun's  limb  can 
be  quite  accurately  fixed  by  observation. 

To  compare  longitudes  by  a  solar  eclipse,  the  observations 
made  on  the  beginning  and  end  at  a  given  place  are  used  as 
means  of  calculating  the  time  of  conjunction — that  is,  the  time 
when  the  sun  and  moon  are  in  the  same  secondary  of  the 
ecliptic.  But  that  event  occurs  at  a  certain  absolute  instant. 
This  computation  being  made  for  each  place,  the  time  of  con- 
junction ought  to  be  exactly  the  same,  so  that  the  difference  in 
the  results  is  wholly  due  to  a  difference  in  the  longitude  of  the 
places.  This  method  of  obtaining  the  longitude  of  a  place  is 
accurate,  but  laborious. 

Occupations  of  stars  by  the  moon  are  much  more  frequent 
than  the  occultation  of  the  sun  •  and  these  are  phenomena  of 


LONGITUDE   BY   THE   TELEGRAPH.  127 

the  same  general  character,  and  may  be  used  in  the  same  way 
for  finding  the  longitude  of  a  place. 

237.  Longitude  ~by  eclipses  of  Jupiter's  satellites. — The  sat- 
ellites of  Jupiter  fall  into  the  shadow  of  that  planet,  as  the 
moon  does  into  the  shadow  of  the  earth.     Every  such  eclipse 
occurs  at  a  certain  time  ;  and  all  who  see  it,  see  it  at  the  same 
instant.     Hence,  these  eclipses  are  favorable  for  determining 
longitudes.      Moreover,  they  are  occurring  every  day,  while 
eclipses  of  the  sun  and  moon  are  rare. 

But,  on  account  of  the  penumbra  of  the  planet,  and  the  con- 
siderable diameter  of  the  satellites,  they  disappear  and  reappear 
gradually.  There  is  difficulty,  therefore,  in  observing  accu- 
rately the  beginning  and  end  of  these  eclipses.  In  order  to 
obtain  the  best  results,  the  telescopes  used  by  different  ob- 
servers ought  to  be  alike  in  aperture  and  power. 

238.  Longitude  ~by  the  lunar  method. — This  is  a  method 
particularly  useful  to  navigators,  because  the  observations  are 
made  by  the  sextant.     It  consists  in  measuring  the  angular 
distance  between  the  moon  and   some   conspicuous   heavenly 
body,  as  the  sun,  or  a  large  planet  or  star,  and  then  correct- 
ing the  observation  for  parallax  and  refraction,  so  as  to  have 
the  true  distance  between  the  bodies,  as  seen  from  the  center 
of  the  earth.      The  observer   must    also   note   the  local  time 
when  this  measurement  is  made. 

Having  with  him  the  Nautical  Almanac,  in  which  the  dis- 
tances, as  seen  from  the  earth's  center,  are  predicted  for  every 
day  and  hour  of  Greenwich  time,  he  looks  for  the  Greenwich 
time  at  which  the  distance  agrees  with  the  distance  as  he  has 
obtained  it.  The  absolute  time  is  the  same ;  hence,  the  dif- 
ference of  time  shows  his  longitude  from  Greenwich. 

The  bodies,  whose  angular  distances  from  the  moon  the 
Kautical  Almanac  gives  for  every  three  hours,  with  propor- 
tional numbers  for  interpolation,  are  the  sun,  Yenus,  Mars, 
Jupiter,  Saturn,  and  nine  bright  fixed  stars. 

239.  Longitude  hy  the  telegraph. — Since  the  invention  of 
the  magnetic  telegraph,  it  has  been  employed  to  determine  the 


128  VELOCITY   OF    ELECTRIC    CUKRENT. 

differences  of  longitude  between  fixed  stations  on  land  with  a 
precision  which  was  before  altogether  unattainable.  Suppose 
two  stations  to  be  connected  by  the  telegraphic  line,  and  that 
there  is  at  each  a  clock  keeping  the  local  time.  The  observ- 
ers agree  beforehand  at  what  time,  by  his  own  clock,  the  one 
at  the  most  easterly  station  shall  commence  giving  signals; 
and  also  at  what  time  the  other  shall  commence  giving  another 
series  according  to  his  clock.  The  interval  between  successive 
signals  is  also  previously  determined.  When  the  moment  ar- 
rives, the  first  observer  strikes  the  telegraphic  key  at  the  ex- 
act beat  of  the  clock,  and  the  second  observer  records  the 
time  of  the  signal  as  shown  by  his  own  clock ;  and  thus  they 
continue  to  do  till  the  full  series  is  recorded.  The  second  ob- 
server then  commences  sending  signals,  which  are  in  like 
manner  recorded  by  the  first.  The  velocity  of  the  electric 
current  is  so  great,  that  the  absolute  time  of  making  a  signal 
at  one  station,  and  of  perceiving  it  at  the  other,  may  be  con- 
sidered identical;  so  that  the  difference  which  is  indicated 
by  the  two  clocks  in  each  case  is  wholly  due  to  difference 
of  longitude.  Still  greater  precision  is  attained  by  causing 
the  signal  key  at  each  station  to  record  its  own  movement  on 
the  line  of  second-marks  made  by  the  clock  at  the  other  sta- 
tion (Art.  46). 

240.  Velocity  of  the  electric  current. — The  method  just  de- 
scribed is  susceptible  of  such  accuracy,  that  it  has  led  to  the 
discovery  of  the  velocity  of  the  current.     For,  if  the  moment 
of  its  arrival  at  the  distant  station  is  not  identical  with  that 
of  the  signal  given,  it  will  indicate  a  difference  of  longitude 
less  than  the  true  difference  when  sent  westward,  but  greater 
than  the  true  difference  when  sent  eastward.     By  this  discrep- 
ancy, if  it  is  appreciable,  the  velocity  of  the  current  becomes 
known.     It  is  found  to  be  about  16,000  miles  per  second. 

241.  Change  of  days  in  circumnavigating  the  earth. — While 
a  person  travels  westward,  he  lengthens  his  days  by  one  hour 
for  every  15°,  or  4  minutes  for  every  degree,  since  he  moves 
along  with  the  apparent  diurnal  motion  of  the  sun.     In  travel- 
ing eastward,  on  the  contrary,  he  shortens  the  days  at  the  same 


DAYS   IN   THE   PACIFIC   OCEAN.  129 

rate,  by  moving  in  opposition  to  the  sun's  daily  progress.  If 
we  suppose  him  to  go  westward  entirely  round  the  earth  to  the 
same  meridian  again,  whether  he  takes  a  longer  or  a  shorter 
time  for  the  journey,  lie  will  lengthen  the  individual  days  suf- 
ficiently to  make  the  whole  number  just  one  day  less  than  if  he 
had  remained  where  he  was.  The  5th  of  a  month  is  to  him  the 
4th ;  and  Tuesday,  according  to  his  reckoning,  is  Monday. 
The  reason  is  obvious ;  for  during  his  journey,  the  earth  has 
made  a  certain  number  of  diurnal  revolutions  from  west  to 
east ;  but  he,  by  going  round  from  east  to  west,  has,  in  respect 
to  himself,  diminished  that  number  by  one. 

All  this  is  exactly  reversed  when  one  goes  round  the  globe 
from  west  to  east.  He  gains  just  a  day  by  making  all  the  days 
of  his  travel  a  little  shorter.  It  is  plain  that  he  makes  one 
more  diurnal  revolution  from  west  to  east  than  the  earth 
does. 

Of  course,  if  these  two  individuals  meet  at  their  place  of 
starting,  they  differ  from  each  other  just  two  days  in  their 
reckoning. 

242.  Ambiguity  as  to  days  among  the  islands  of  the  Pacific 
Ocean. — If  an  island  in  the  Pacific  were  settled  by  navigators, 
who  had  gone  westward  around  Cape  Horn,  and  also  by  others, 
who  had  sailed  eastward  around  the  Cape  of  Good  Hope,  the 
reckoning  of  these  two  parties  would  differ  by  one  day.  To 
the  former,  a  day  will  be  the  first  of  a  month  when  it  is  the  2d 
to  the  latter.  It  is,  in  fact,  true  that  there  are  islands  lying 
contiguous  to  each  other  which  have  this  difference  of  reckon- 
ing. 

If  inhabited  land  extended  entirely  round  the  earth,  it  would 
be  necessary  to  fix  arbitrarily  on  some  meridian  on  which  the 
change  of  day  should  be  made.  For  it  is  impossible  that  the 
reckoning  of  days  should  go  on  unbroken  around  the  earth. 
The  arbitrary  meridian  would  separate  between  places  which 
differ  a  day  from  each  other ;  so  that,  on  the  west  side  of  it, 
the  time  is  one  day  later,  both  in  the  month  and  the  week,, 
than  on  the  east  side. 


130 


WATEE  ACTED   ON   BY   THE   MOON. 


CHAPTER  XIV. 

THE   TIDES. 

243.  Definitions. — The  tides  are  the  daily  rising  and  fall- 
ing of  the  waters  of  the  ocean.     When  the  water,  in  this  daily 
oscillation,  has  reached   its  highest  point,  it  is   called  high- 
water  •  at  its  lowest  point,  it  is  called  low-water.     While  the 
water  is  rising,  it  is  called  j$6^;  and  while  falling,  ebl. 

A  lunar  day  is  the  time  between  two  successive  culmina- 
tions of  the  moon.  Its  length  is  about  24h.  52m.,  being  nearly 
an  hour  longer  than  a  solar  day  on  account  of  the  rapid  east- 
ward motion  of  the  moon.  The  tides  make  their  revolutions 
within  the  lunar  day. 

Twice  in  a  lunation  high-water  is  at  a  maximum,  and  twice 
it  is  at  a  minimum;  the  former  are  called  spring  tides,  the 
latter,  neap  tides.  The  spring  tides  occur  near  the  time  of 
syzygies,  the  neap  tides  near  the  time  of  quadratures. 

244.  Opposite  tides. — There  are  two  tide-waves  on  opposite 
sides  of  the  globe,  moving  around  it  from  east  to  west,  and  ar- 
riving at  any  place  at  intervals,  whose  mean  value  is  12h. 
26m.,  or  half  a  lunar  day.     Since  the  mean  diurnal  motion  of 
each  of  the  two  opposite  tides  is  the  same  as  that  of  the  moon, 
the  action  of  the  moon  must  be  regarded  as  the  principal  cause 
of  the  tides. 

245.  Form  of  the  water  acted  on  R-  64- 
by  the  moon. — If  the  earth  were  cov- 
ered with  water,  and  no  force  were 

exerted  except  gravitation  toward  the 
earth  itself,  its  form  would  be  exactly 
spherical,  as  represented  in  Fig.  64. 
But  if  a  distant  body,  as  the  moon, 
should  also  attract  it,  the  sphere  would 
be  changed  into  a  prolate  spheroid — 
that  is,  into  a  form  produced  by  re- 
volving an  ellipse  about  its  major  axis.  Let  the  moon  be  in 


JOINT   ACTION   OF   THE    SUN  AND    MOON.  131 

Jie  direction  of  CE  produced,  and  suppose  the  center  of  gravity 
of  the  nearer  half  of  the  water,  DEF,  to  be  at  A,  and  that  of 
the  remote  half  at  B,  while  the  center  of  the  earth,  as  a  whole, 
is  at  C.  Since  A  is  more  attracted  than  C,  and  C  more  than 
B,  the  form  of  equilibrium  must  be  disturbed,  and  some  of  the 
water  will  flow  toward  E,  and  other  parts  toward  G,  till  the 
particles  are  in  equilibrio  between  their  unequal  tendencies  to 
the  moon,  and  their  gravity  on  the  inclined  surface  of  the 
spheroid.  E  and  G  are  the  highest  points  of  the  spheroid,  and 
all  points  on  the  circle  DF  (perpendicular  to  EG)  are  the 
lowest.  Every  section  through  EG  is  an  ellipse,  whose  major 
axis  is  EG,  and  whose  minor  axis  is  equal  to  DF.  The  ellip- 
ticity  of  the  section  will  obviously  depend  not  only  on  the 
strength  of  the  moon's  attraction,  but  also  on  the  difference  be- 
tween the  attractions  on  the  nearer  and  remoter  parts. 

In  the  case  of  the  earth  and  moon,  it  is  computed  that  the 
major  axis  would  exceed  the  minor  by  5  feet — that  is,  the  tides 
would  be  only  2-J-  feet  high,  and  on  opposite  sides  of  the 
earth,  one  directed  toward  the  moon,  the  other  from  it.  The 
tide  on  the  side  nearest  the  moon  is  sometimes  called  the  direct 
tide ;  the  one  on  the  remote  side,  the  opposite  tide. 

246.  Tides  by  the  sun. — The  same  kind  of  effect  is  pro- 
duced by  the  sun  as  by  the  moon.     But  the  distance  of  the  sun 
is  so  great,  that  though  it  attracts  the  earth  more  than  the 
moon  does,  yet  the  difference  of  its  attractions  on  the  several 
parts  is  less.     The  power  of  the  moon  to  raise  a  tide  is  to  that 
of  the  sun  about  as  5  to  2. 

247.  Joint  action  of  the  sun  and  moon. — At  the  time  of 
conjunction,  the  moon  and  sun  attract  in  the  same  direction, 
and  therefore  the  tides  are  equal  to  the  sum  of  the  lunar  and 
solar  tides.     The  same  is  true  at  opposition,  because  each  body 
produces  two  tides  at  once ;  and  the  direct  lunar  tide  coincides 
with  the  opposite  solar  tide,  and  vice  versa.     These  are  the 
spring  tides  which  occur  at  the  syzygies. 

At  quadratures,  each  body  raises  a  tide  at  the  expense  of 
that  raised  by  the  other.  For  if  the  moon  is  in  the  direction 
of  EG  produced  (Fig.  64),  it  causes  high-water  at  E  and  G, 


132 


DIUKNAL   INEQUALITY. 


and  low- water  at  D  and  F.  And  if  the  sun  is  in  the  direction 
of  DF  produced,  it  causes  high-water  at  D  and  F,  and  low- 
water  at  E  and  G.  As  the  lunar  tides  are  the  highest,  E  and 
G  are  the  neap  tides,  made  less  by  this  action  of  the  sun,  than 
if  the  moon  had  acted  alone. 

248.  Effect  of  the  inertia  of  water. — Tf  the  moon  and  earth 
were  at  rest,  the  tides  would  be  directed  exactly  to  and  from 
the  moon.     But  while  the  waters  are  flowing  toward  these 
points,  the  moon,  by  the  diurnal  motion,  passes  westward,  and 
causes  them  to  change  the  places  at  which  they  tend  to  accu- 
mulate.    Thus,  even  if  the  wave  were  unchecked  by  the  shores 
of  continents  and  islands,  the  summit  would  be  two  or  three 
hours  behind  the  moon  in  passing  a  given  meridian. 

249.  Diurnal  inequality. — At  a  given  place,  the  two  tides 
which  follow  the  culmination  of  the  moon  will  vary  in  height, 
according  to  the  relation  between  the  latitude  of  the  place  and 
the  moon's  declination.     If  the  moon,  M  (Fig.  65),  is  on  the 
equator,  it  is  clear  that  the  tides  on  the  equator,  EQ,  are  great- 
est, and  that  in  other  places  they  are  less,  as  the  latitude  is 
greater.     But  the  two  successive  tides  at  any  place  are  equal ; 
for,  by  the  rotation  on  NS,  the  tide  at  B  in  12  J  hours  will 
come  round  to  A,  and  be  equal  to  the  tide  now  there.     The 
same  is  true  of  the  tides  C  and  D,  or  F  and  G.     Hence,  when 
the  moon  has  no  declination,  there  is  no  diurnal  inequality. 


But  suppose  the  moon  has  a  northern  declination,  as  in  Fig. 
66.     Then  the  highest  points  of  the  tide  are  at  A  in  north  lat- 


COTIDAL   LINES. 


133 


itude,  and  D  in  south.  At  A,  where  the  direct  tide  is  large, 
the  opposite  tide  now  at  B  will  arrive  in  12^  hours,  and  will 
be  small.  But  at  C,  this  is  reversed ;  the  direct  tide  is  small, 
and  the  opposite  one  (now  at  D,  and  arriving  at  C  12^  hours 
later),  is  large.  Therefore,  when  the  declination  and  the  lat- 
itude are  both  north,  or  both  south,  the  direct  tide — that  is,  the 
tide  which  first  succeeds  the  upper  culmination  of  the  moon — 
is  larger  than  the  opposite  tide ;  but  if  one  is  north,  and  the 
other  south,  the  direct  tide  is  smaller  than  the  opposite  tide. 
This  difference  in  the  height  of  the  two  successive  tides  is 
called  the  diurnal  inequality. 

25O.  Change  of  direction  and  velocity  caused  by  coasts. — 
The  tide-wave,  which  would  move  regularly  westward  around 
the  earth,  if  it  were  wholly  covered  by  deep  water,  is  exceed- 
ingly broken  up  and  changed,  both  in  direction  and  velocity, 
by  coasts  and  shoals.  Its  general  direction  is  westward ;  but 
as  it  can  pass  the  continents  only  at  their  southern  extremities, 
it  bears  to  the  northwest,  and  then  to  the  north,  in  the  Atlan- 
tic and  Pacific  oceans ;  and  when  it  enters  seas  or  channels,  it 
usually  bends  its  course  in  the  direction  of  their  length. 


Fig.  67.. 


251.  Cotidal  lines. — These  are  lines  drawn  on  a  chart  of 
the  oceans,  showing  the  posi- 
tion of  the  summit  of  the  tide- 
wave  for  each  hour  of  a  day. 
Such  a  system  of  lines  expresses 
to  the  eye  the  direction  and  ve- 
locity of  the  tide  at  all  places. 
Thus,  on  the  open  ocean,  the 
figures  1,  2,  3,  4  (Fig.  67)  show 
the  situation  of  one  and  the 
same  tide- wave  at  those  hours, 
respectively.  And  in  the  chan- 
nel which  extends  northward, 
the  wave,  having  separated  from 
the  ocean  tide,  advances  north- 
ward, and  occupies  the  places 
marked  at  the  hours  indicated.  The  wave  advances  most  rap- 


ESTABLISHMENT  OF   A   POET. 

idly  in  the  deepest  water.  Hence,  the  front  i?  generally  convex, 
as  in  Fig.  67,  since  it  moves  fastest  in  the  central  part,  where 
the  water  is  deepest.  For  this  reason,  also,  the  tide  may  occu- 
py as  long  a  time  in  running  through  a  long  channel  of  shal- 
low water  as  in  advancing  half  round  the  earth.  The  greatest 
velocity  of  the  tide  in  the  deep,  open  ocean,  is  near  1,000  miles 
per  hour.  Some  channels  are  affected  by  tides  entering  at  both 
extremities.  For  example,  the  German  Ocean  and  English 
Channel  receive  the  Atlantic  tide  both  at  the  north  and  at  the 
south  end.  As  a  consequence,  the  tide  system  is  doubled, 
causing,  at  some  points,  four  tides  per  day. 

252.  Modification   in   the   height   of  the    tide   caused  by 
coasts. — The  relation  of  coast  lines  to  each  other  also  very 
much  affects  the  height  of  the  tide  at  particular  places.    When 
the  tide  directly  enters  a  broad-mouthed  bay,  it  grows  higher 
as  the  bay  contracts  in  breadth ;  and  at  the  head  of  the  bay, 
there  is  usually  found  the  greatest  height  of  all.     One  of  the 
most  remarkable  examples  is  the  Bay  of  Fundy.     The  western 
extremity  of  the  Atlantic  tide-wave,  after  entering  this  bay,  is 
gradually  contracted  by  the  shores  as  it  advances,  till,  at  the 
head  of  the  bay,  it  sometimes  rises  to  70  feet. 

The  height  of  the  tide  on  the  coast  is  generally  greater  than  in 
the  open  ocean,  owing  to  the  effect  of  shoal  water.  The  most 
advanced  part  of  the  wave  moves  slower  than  the  hinder  por- 
tion ;  so  that  the  cross-section  of  the  ridge  becomes  shorter, 
and  therefore  higher,  as  the  depth  of  water  diminishes. 

The  mean  height  of  the  spring  tides  at  any  place  is  called 
the  unit  of  altitude  for  that  place. 

253.  Establishment  of  a  port. — This  phrase  signifies  the 
mean  interval  between  the  culmination  of  the  moon  and  the 
arrival  of  the  tide  at  a  given  place.     At  every  meridian,  the 
tide  arrives  later  than  the  body  which  causes  it ;  but  the  delay 
varies  exceedingly  at  different  localities,  011  account  of  shoal 
water,  direction  and  length  of  channel,  etc.     Even  at  the  same 
place,  the  delay  during  a  lunation  varies   according  as  the 
small  solar  tide  precedes  or  follows  the  large  lunar  one;  for 
the   summit  lies  between  them.     It  is  the  mean  interval  at 


TIDES   MODIFIED   BY   DISTANCE.  135 

a   given   port,   which,  is    called    the    establishment    of   that 
port. 

254.  Tides  of  lakes  and  inland  seas. — In  general,  the  tides 
of  lakes  and  inland  seas  are  scarcely  perceptible.     The  reason 
is,  their  extent  is  so  small,  that  all  parts  are  to  be  considered  as 
almost  equidistant  from  the  moon.     There  is  little  opportunity 
for  water  to  be  attracted  from  the  more  distant  to  the  nearer 
part.     The  largest  North  American  lakes  have  tides  but  an 
inch  or  two  in  height.     In  the  Mediterranean,  however,  which 
derives  no  tide  from  the  ocean,  the  tide-wave  reaches  1J  or  2 
feet. 

255.  Tides  modified  hy  the  surfs  and  'moon's  change  of  dis- 
tance.— The  difference  of  the  moon's  attraction  on  the  several 
parts  of  the  earth  is  greatest  when  the  moon  is  nearest,  and 
least  when  it  is  most  distant.     The  same  is  true  of  the  sun. 
Hence,  the  tides  of  each  sidereal  month  have  a  periodical  in- 
crease and  decrease  as  the  moon  passes  through  its  perigee  and 
apogee.     They  have  a  like,  though  much  smaller,  change  each 
year,  at  the  perihelion  and  aphelion  of  the  earth's  orbit.     By 
the  revolution  of  the  apsides  of  the  moon's  orbit,  these  maxima 
and  minima  will  alternately  coincide  once  in  9  years.     Com- 
bining these  changes  with  those  at  syzygy  and  quadrature,  the 
height  of  the  greatest  possible  spring  tide,  to  that  of  the  least 
possible  neap  tide,  is  as  10  to  3. 


136  CLASSIFICATION   OF   THE    PLANETS. 


CHAPTER  XY. 

THE    PLANETS. — TABULAR   STATEMENTS. — MERCURY. — VENUS. 

MARS. 

256.  Names  and  classification  of  the  planets. — The  planets 
are  solid  spherical  bodies  revolving  about  the  sun  in  orbits 
which  are   nearly  circular.      The  name  "  planet"  signifies  a 
wanderer,  and  was  given  to  these   bodies  because  they  con- 
tinually change  their  places  among  the  fixed  stars,  generally 
moving  from  west  to  east,  but  sometimes  from  east  to  west. 
These  apparently  irregular  motions  are  fully  explained  by  our 
own  annual  motion,  the  earth  on  which  we  live  being  one  of 
the  planets. 

The  planets  are  naturally  arranged  in  three  classes. 

1.  Four  small  planets  near  the  sun,  of  which  the  earth  is  the 
largest — namely,  Mercury,  Venus,  Earth,  Mars. 

2.  The  planetoids,  an  indefinite  number  of  bodies,  too  small 
to  be  measured  with  certainty,  and  occupying  a  ring  outside 
of  the  first  class.     They  are  also  called  asteroids,  and  minor 
planets. 

3.  Four  large  planets,  moving  outside  of  the  ring  of  plan- 
etoids, widely  separated  from  each  other,  and  at  vast  distances 
from  the  sun.     These  are  Jupiter,  Saturn,  Uranus,  Neptune. 

Two  planets  of  the  first  class,  Mercury  and  Yen  us,  revolve 
in  orbits  within  the  earth's  orbit.  These  are  called  inferior 
planets,  being  lower  down  in  the  solar  system  than  the  earth 
is.  All  the  others,  including  the  planetoids,  are  called  superior 
planets ;  because,  in  relation  to  the  sun,  the  great  center  of  at- 
traction, they  are  higher  than  the  earth,  and  revolve  in  orbits 
exterior  to  the  earth's  orbit. 

257.  Satellites. — There  is  another  class  of  spherical  bodies, 
holding  a  subordinate  place  in  the  solar  system,  since  they  re- 
volve around  the  planets  as  centers.     These  are  called  satel- 
lites.    The  moon,  already  described  in  Chapter  X.,  is  a  satellite 


PERIODIC   TIMES   OF   PLANETS. 


137 


of  the  earth.  They  are  distributed  as  follows :  the  earth  has 
1;  Jupiter,  4;  Saturn,  8 ;  Uranus,  4  ;  Neptune,  1.  Mercury, 
Yenus,  and  Mars  have  no  satellites. 

The  satellites  are  also  called  secondary  planets ;  and  the 
planets,  in  distinction  from  them,  primary  planets. 

258.  Distances  of  the  planets  from  the  sun. — The  follow- 
ing table  presents  the  mean  distances  of  the  planets  from  the 
sun  in  millions  of  miles,  and  also  their  relative  distances,  the 
earth's  being  called  1. 


Mean  Distances. 

Relative 
Distances. 

Mercury 

37,000,000 

0  39 

Venus  

69,000,000 

0.72 

Earth  

95,000,000 

1.00 

Mars  . 

145  000  000 

1  52  ' 

Planetoids  

254,000,000 

2.67 

Jupiter. 

496,000,000 

5.20 

Saturn  .  .  . 

909,000,000 

9.54 

Uranus  

1,828,000,000 

19.18 

Neptune  

2,862,000,000 

30.04 

It  appears  by  this  table,  that  the  remotest  planet  is  77  times 
as  far  from  the  sun  as  the  nearest.  Hence  it  is  that  orreries, 
unless  of  inconvenient  size,  always  fail  of  truly  representing 
the  planetary  distances.  The  same  is  generally  true  of  dia- 
grams. 

259.  Periodic  times  of  the  planets. — The  following  table 
contains  the  length  of  the  sidereal  revolutions  in  months  and 
years,  which  is  the  most  convenient  form  for  the  memory ; 
their  length  in  days  and  decimals,  for  calculations ;  their  mean 
daily  motion ;  and  the  time  of  their  diurnal  rotations,  so  far  as 
known,  in  hours  and  decimals. 


138 


MAGNITUDES. 
II. 


Sidereal  Revolu- 
tion. 

Sidereal  Revo- 
lution in  Days. 

Mean  Daily 
Motion. 

Diurnal 
Rotation. 

Mercury  .... 
Arenns  

3  months. 
71     « 

8W69 

224.701 

4°    5/32//.6 
1°  36'    7"  7 

24.91  h. 
23  35  " 

Earth  

1    year 

305.256 

0°  50'    s"  3 

24  00  " 

Mars         .  . 

2       u 

6&6  980 

0°  31'  26  ''  5 

24  66  " 

Planetoids  .  . 
Jupiter 

4J     « 
12       " 

4332  585 

0°    4'  59"  1 

9  92  " 

Saturn. 

29       " 

10759  220 

0°    2'    0"  5 

10  48  " 

Uranus  

84       « 

30686.821 

0°    0'  42".2 

Neptune.  .  .  . 

164       « 

60126.722 

0°    0'  21".6 

It  will  be  found,  by  comparing  the  squares  of  any  two  periods 
in  Table  II,  and  the  cubes  of  the  corresponding  distances  in 
Table  I,  that  their  ratios  are  nearly  the  same ;  and  this  should  be 
true  according  to  Kepler's  third  law  (Art.  1 19).  Thus,  for  Nep- 
tune and  the  earth,  303  :  I3  =  27,000;  and  1642  :  I2  =  26,896. 
So  also,  while  Neptune  is  77  times  as  far  from  the  sun  as  Mer- 
cury is,  its  period  of  revolution  is  683  times  as  long.  For, 
77s  :  I3  : :  6832  :  I2,  nearly. 

Since  the  periods  increase  more  rapidly  than  the  radii  of  the 
orbits,  the  velocities  of  the  planets  must  become  less,  the  fur- 
ther they  are  from  the  sun.  The  distance  described  by  Mer- 
cury in  a  day  is  nearly  nine  times  that  which  Neptune  passes 
over  in  the  samo  time. 

III. 


Diameters. 

Apparent 
Diameters. 

Volumes. 

Sun  

887,000 

32' 

1.400,000 

Mercury       

2,950 

0'     8" 

rV 

Venus  

7,800 

0'  17" 

& 

Earth          

7,912 

Mars           

4,500 

0'     6" 

f 

Jupiter  

89,000 

V  37" 

1,400 

Saturn  

79,000 

0'  16" 

1,000 

Uranus         .  . 

35,000 

0'     4" 

86 

Neptune  

31,000 

0'     2" 

60 

MAGNITUDES. 


139 


2GO.  Magnitudes  of  the  planets. — Table  III  gives  the  di- 
ameters of  the  sun  and  planets  in  miles,  their  mean  apparent 
diameters,  and  their  volumes  compared  with  the  earth. 

In  comparing  the  numbers  of  this  table,  it  is  noticeable  that 
the  four  large  planets  dimmish  in  size  as  they  are  more  distant 
from  the  planetoids.  The  same  would  be  true  of  the  four 
small  planets,  if  only  Mars  were  placed  between  Yenus  and 
Mercury. 

It  is  a  singular  fact,  also,  that  the  diameters  of  the  large 
planets  beyond  the  planetoids  are  not  far  from  ten  times  as 
large,  respectively,  as  those  of  the  small  ones  within  that  group. 


Diam.  of  Jupiter 
"       Saturn 
"       Uranus 
"       Neptune 
and  the  sum 


that  of  the  earth  :  :  11 

"       Yenus      :  :  10 

"       Mars        :  :    8 

"       Mercury  :  :  10 

the  sum    :  :  10 


1,  nearly. 

" 


Another  remarkable  fact  appears  on  comparing  the  diameters 
in  Table  III,  and  the  times  of  diurnal  rotation  in  Table  II. 
The  four  small  planets  all  rotate  in  periods  of  about  24  hours. 
But  the  large  planets,  so  far  as  known,  revolve  in  about  10 
hours.  Hence,  the  equatorial  velocity  of  rotation  is  far  greater 
on  the  large  than  on  the  small  planets.  That  on  Jupiter,  for 
example,  is  27  times  as  great  as  that  on  the  earth. 

The  dimensions  of  the  planetoids  are  not  given  in  the  table, 
being1  too  small  for  measurement.  One  or  two  of  the  largest 
are  thought  to  be  from  100  to  200  miles  in  diameter. 

IY. 


Masses. 

Density. 

Specific 
Gravity. 

Sun 

354,000. 

0.25 

1.4 

Mercury         .  .    . 

0.12 

2.01 

11.0 

Yenus 

0.88 

0.97 

5.3 

Earth 

1.00 

1.00 

5.5 

Mars  .... 

0.13 

O.T2 

3.9 

Jupiter.  . 

338.03 

0.24 

1.3 

Saturn  

101.06 

0.13 

0.7 

Uranus  

14.79 

0.15 

0.8 

Neptune  . 

24.65 

0.27 

1.5 

140  DIAMETERS   OF   PLANETS. 

261.  M asses  and  densities  of  the  planets. — Table  TV  ex- 
hibits the  masses  and  densities  of  the  sun  and  planets,  the 
earth  being  called  1 ;  also  their  specific  gravities. 

It  appears  from  table  IY,  that  the  small  planets  are  much 
more  dense  than  the  large  planets  and  the  sun. 

262.  The  sun  and  planets  compared. — By  Table  III,  we  see 
that  the  sun  has  10  times  the  diameter,  and  1,000  times  the 
volume  of  Jupiter,  the  largest  planet  in  the  system.     Table  IY 
shows  that  the  mass  of  the  sun  is  also  more  than  1 ,000  times  as 
great  as  that  of  Jupiter,  and  TOO  times  greater  than  the  united 
masses  of  all  the  planets.     Its  attraction  mainly  controls  the 
movements  of  all  the  planets,  satellites,  and  comets.     Hence, 
these  bodies  describe  their  various  paths  about  it,  scarcely  dis- 
turbing it  from  a  state  of  rest.     For  this  reason,  this  system  of 
bodies  is  called  the  solar  system,. 

263.  Diameters  of  planets,  and  their  distances  from  the 
sun. — One  of  the  most  remarkable  facts  relating  to  the  planets 
is  brought  to  view  in  comparing  the  distances  in  Table  I  with 
the  diameters  in  Table  III.    While  the  diameters  of  the  planets 
are  only  a  few  thousands  of  miles,  their  distances  from  the  sun 
are  many  millions.     The  diameter  of  Neptune's  orbit  is  more 
than  20,000  times  the  diameters  of  all  the  planets  added  to- 
gether.    To  attempt  to  represent  both  the  distances  and  mag- 
nitudes of  the  planets  in  their  proportions,  by  an  orrery  or 
diagram,  is  out  of  the  question.* 

264.  Directions  of  the  planetary  motions. — It  has  been 

*  A  diagram  of  the  planetary  system,  which  is  most  useful  in  the  lecture- 
room,  is  prepared  as  follows  : 

Stretch  upon  the  wall  a  piece  of  black  cambric,  as  long  as  the  room  will 
allow,  say  30  feet.  This  length  may  be  taken  for  the  radius  of  Neptune's 
orbit.  At  one  end,  attach  a  circle  of  white  cloth,  \  inch  in  diameter,  for  the 
sun.  From  it,  as  a  center,  describe  arcs  across  the  cloth,  at  proper  distances 
for  the  several  orbits,  and  sew  white  tape  on  these  arcs,  to  render  them  visible. 
These  white  lines  represent  short  portions  of  the  orbits.  We  then  have  the 
planetary  distances  and  the  size  of  the  sun  upon  one  scale.  There  should  be 
no  attempt  to  represent  the  planets  themselves,  since  Jupiter,  the  largest 
would  be  only  »/40  of  an  inch  in  diameter,  and  none  of  them,  if  of  proper  size, 
would  be  visible  across  the  room.  The  cloth  can  be  attached  to  rollers  at 
the  ends,  and  conveniently  rolled  up  when  not  in  use. 


APPARENT  MOTIONS  OF   MERCURY. 


141 


stated  in  preceding  chapters  that  all  the  motions  of  the  sun, 
earth,  and  moon  are  from  west  to  east.  The  same  thing  is 
true,  in  general,  of  all  the  planets  and  satellites ;  and  in  nearly 
every  case  the  inclination  to  the  ecliptic  is  very  small.  The 
only  exceptions  are  found  in  the  satellites  of  Uranus  and  Nep- 
tune, whose  planes  of  revolution  are  nearly  perpendicular  to 
the  ecliptic,  and  the  motion  in  them  from  east  to  west.  All 
the  planetoids  yet  discovered  revolve  from  west  to  east,  though 
the  orbit  of  one  of  them  has  an  inclination  as  large  as  34°. 

Since  the  motions  in  the  solar  system  are  so  generally  from 
west  to  east,  this  is  regarded  as  direct  motion ;  and  any  mo- 
tions, real  or  apparent,  which  are  from  east  to  west,  are  called 
retrograde. 

MERCURY. 

265.  Tabular  statements. — Mean  distance  from  the  sun, 
37,000,000  miles;  periodic  time,  3  months;  diameter,  2,950 
miles;  diurnal  rotation,  2491  hours;  specific  gravity,  11.0. 

Fig.  68. 


266.  Apparent  motions. — Mercury  is  an  inferior  planet, 
whose  orbit  is  far  within  the  earth's  ;  for  it  is  seen  alternately 
east  and  west  of  the  sun,  and  never  more  than  29°  from  it. 
Let  E  (Fig.  68)  be  the  earth,  supposed,  for  the  present,  to  be  at 


142 


APPARENT   MOTIONS   OF   MERCUKY. 


rest ;  the  circle  ABD,  the  orbit  of  Mercury ;  S,  the  sun  ;  and 
IT  A7,  the  sky,  on  which  the  bodies  are  seen  projected.  When 
Mercury  is  at  B,  it  is  seen  at  B' ;  as  it  passes  through  D  to  A, 
it  appears  to  advance  to  A!  ;  as  it  is  now  coming  toward  the 
earth,  it  seems  to  be  stationary  at  A!  ;  then  from  A  through  C 
to  B,  it  appears  to  retrograde  from  A'  to  B',  where  it  is  again 
stationary,  as  it  moves  away  from  us.  Since  the  sun  appears  at 
S',  the  planet  passes  by  it,  both  when  advancing  and  when 
retrograding. 

When  the  planet  is  at  D  and  C,  it  is  in  conjunction  with  the 
sun  ;  at  C,  between  the  earth  and  sun,  it  is  said  to  be  in  the 
inferior  conjunction  ;  at  D,  in  superior  conjunction.  B  and  A 
are  called  the  points  of  greatest  elongation.  At  superior  con- 
junction, the  motion  of  Mercury  appears  to  be  forward  ;  at  the 
inferior  conjunction,  backward ;  and  if  the  earth  were  at  rest, 
as  we  are  now  supposing,  the  planet  would  appear  stationary 
at  the  points  of  greatest  elongation. 

Fig.  69. 


D 


267.  The  motions  of  Mercury  as  modified  ly  the  earth's 
motion.— To  simplify  the  case,  it  was  supposed,  in  the  preced- 
ing article,  that  the  earth  is  at  rest.  But  the  earth  moves  in 


SYNODIC AL   PERIOD    OF   MERCURY.  143 

nearly  the  same  direction  as  Mercury,  making  about  one  rev- 
olution while  Mercury  makes  four  (Table  II).  The  effect  is  to 
lengthen  the  arc  of  apparent  advance,  and  shorten  that  of  re- 
trogradation.  Thus,  let  the  earth  be  at  A  (Fig.  69),  when 
Mercury  is  at  F  ;  then  it  will  appear  in  the  sky  at  L.  While 
the  earth  is  advancing  to  B,  Mercury  passes  the  inferior  con- 
junction, and  arrives  at  G,  and  appears  at  M,  having  moved 
apparently  backward  from  L  to  M.  As  the  earth  moves  to  C, 
Mercury  describes  GKH,  and  is  at  superior  conjunction  ~N. 
Again,  while  the  earth  moves  to  D,  Mercury  passes  round  to 
G,  still  advancing  in  the  sky  to  O.  But  while  the  earth  de- 
scribes DE,  Mercury  again  passes  the  inferior  conjunction  from 
G  to  K,  and  apparently  retrogrades  from  O  to  P  ;  after  which, 
it  begins  once  more  to  advance.  Thus,  by  the  earth's  motion, 
the  planet  is  made  to  retrograde  through  a  shorter  arc,  and  to 
advance  through  a  longer  one,  than  if  the  earth  were  at  rest. 

268.  Stationary  points. — If  the  earth  were  at  rest,  as  sup- 
posed in  Fig.  68,  the  points  where  the  planet  would  appear 
stationary,  in  relation  to  the  stars,  would  be  A  and  B,  at  which 
tangents  drawn  from  the  earth  would  meet  the  orbit.     But  the 
earth's  motion  removes  the  apparently  stationary  points  a  little 
way  toward  the  inferior  conjunction.     For,  in  order  to  appear 
stationary,  the  advance  which  the  earth's  motion  causes,  must 
be  just  neutralized  by  the  retrogradation  of  Mercury.     This 
planet  appears  stationary,  when  its  elongation   from  the  sun 
is   15°  or  20°,  according  as  it  is  nearer  the  perihelion  or  the 
aphelion. 

269.  The  synodical  period  of  Mercury. — This  is  the  time 
in  which  it  goes  from  a  conjunction  to  the  next  conjunction  of 
the  same  kind — that  is,  describes  one  revolution  relatively  to 
the  earth  instead  of  a  star. 

The  sidereal  period  having  been  obtained  by  observing  the 
planet's  return  to  its  node,  the  synodical  period  can  be  com- 
puted from  it  by  using  the  relative  motions  of  Mercury  and  the 
earth,  just  as  we  find  the  time  in  which  the  minute-hand  of  a 
watch  will  overtake  the  hour-hand.  The  synodical  period  of 
Mercury  can  also  be  found  independently,  by  means  of  transits 


144:  TRANSITS   OF   MERCURY. 

across  the  sun's  disk.     The  synodical  period  of  Mercury  is  110 
days,  which  is  nearly  a  month  longer  than  its  sidereal  period. 

270.  Form  and  position  of  Mercury's  orbit — The  orbit  of 
Mercury  is  more  eccentric,  and  more  inclined  to  the  ecliptic, 
than  that  of  any  other  of  the  eight  planets.     While  the  eccen- 
tricity of  the  earth's  orbit  is  only  Jff,  that  of  Mercury  is  nearly 
J.     Yet  this  renders  the  minor  only  ^  shorter  than  the  major 
axis ;  so  that  the  form  of  the  most  eccentric  of  the  planetary 
orbits,  if  correctly  drawn,  would  appear  to  the  eye  to  be  a 
circle. 

The  inclination  of  Mercury's  orbit  to  the  plane  of  the  eclip- 
tic is  7°. 

271.  Phases  of  Mercury. — At  the  inferior  conjunction,  C 
(Fig.  f>8),  the  unilluminated  side  of  Mercury  is  turned  toward 
the  earth,  so  that,  like  the  n^w  moon,  it  is  invisible.     At  the 
superior  conjunction,  D,  its  illuminated  side  is  toward  us,  and 
it  is  full.     At  A  or  B,  where  the  ray  AS,  and  our  line  of  vis- 
ion, AE,  are  at  right  angles,  the  phase  is  a  semicircle.     On  the 
arc  ACB  occur  the  crescent  phases  :    on  BDA,  the  gibbous 
phases. 

272.  Point  of  greatest  brightness. — Mercury  is  not  bright- 
est when  full,  because  it  is  then  too  far  distant.     It  is  not 
brightest  when  nearest,  because  its  dark   side  is  toward  us. 
Nor  is  it  brightest  at  the  place  of  greatest  elongation  ;   but 
beyond  it,  toward  the  superior  conjunction,  when  about  22° 
from  the  sun.     Its  apparent  diameter,  when  nearest  the  earth, 
and  when  most  distant  from  it,  is  as  2^  to  1 . 

273.  Transits  of  Mercury. — As  Mercury,  at  the  inferior 
conjunction,  passes  nearly  between  the  earth  and  sun,  it  may 
possibly  come  exactly  in  a  line  with  them,  and  thus  be  seen  as 
a  black  round  spot  going  across  the  sun's  disk.     This  phenom- 
enon is  called  a  transit  of  Mercury.     If  the  plane  of  its  orbit 
were   coincident  with  that  of  the   ecliptic,  a  transit   would 
obviously  occur  at  every  inferior  conjunction.     Since  the  angle 
between  the  two  planes  is  7°,  the  planet  can  not  be  seen  on 


APPARENT   MOTIONS   OF   VENUS.  145 

the  disk,  unless  near  the  node,  for  its  perpendicular  distance 
from  the  ecliptic  must  be  less  than  the  sun's  apparent  semi- 
diameter — that  is,  less  than  16'.  By  a  simple  calculation,  like 
that  in  Art.  203,  it  is  found  that  the  limit  of  transit  for  Mer- 
cury is  2°  10'. 

274.  Node  months  for  Mercury. — The  nodes  of  Mercury's 
orbit   lie   in   that  part  of  the  heavens  which  the  sun  passes 
through  in  May  and  November.     Therefore,  a  transit  of  that 
planet  can  occur  only  in  those  months.     More  transits  happen 
in  November  than  in  May,  because  the  planet  is  nearer  peri- 
helion in  November,  and  therefore  more  likely  to  be  projected 
On  the  sun's  disk.     After  the  lapse  of  ages,  the  months  will 
change,  on  account  of  the  slow  retrograde  motion  of  the  nodes. 

275.  Intervals  "between  transits. — While  the  earth  makes 
13  revolutions  from  a  node  to  the  same  node  again,  Mercury 
makes  54  revolutions,  very  nearly.     Hence,  in  13  years  after  a 
transit,  the  two  bodies  will  return  so  nearly  to  the  same  rela- 
tions to  the  node,  that  another  transit  is  likely  to  occur.     The 
least  interval  between  transits  at  the  same  node  is  7  years,  in 
which  time  Mercury  makes  very  nearly  29  revolutions.     As 
these  are  both  odd  numbers,  the  period  may  be  halved,  and  a 
transit  may  occur  in  3£  years  at  the  other  node.     This  is  the 
shortest  interval.     The  transits  of  Mercury  in  the  last  half  of 
the  present  century  are  the  following:  November  11,1861; 
November  4,1868;  November  7,  1881;   May  9,  1891;   No- 
vember 10,  1894. 

VENUS. 

276.  Tabular  statements. — Mean  distance   from  the  sun, 
69,000,000  miles;  periodic  time,  7i  months;  diameter,  7,800 
miles;  diurnal  rotation,  23.35  hours;  specific  gravity,  5.3. 

277.  Apparent  motions. — Like  Mercury,  Yenus  appears  to 
pass  back  and  forth  by  the  sun,  reaching  a  distance  of  47°  at 
its  greatest  elongation.     This  proves  it  to  be  an  inferior  planet, 
between  Mercury  and  the  earth.    Its  sidereal  period  approaches 
so  near  to  that  of  the  earth,  that  its  synodic  period  is  length- 

10 


146  TRANSITS   OF   VENUS. 

ened  to  nearly  Ij  years.  Hence,  after  making  an  apparent 
retrograde  motion,  as  LM  (Fig.  69),  it  advances  once  and  two- 
tliirds  round  the  heavens  before  it  commences  the  next  retro- 
grade arc,  OP. 

278.  Phases  and    brightness    of    Venus. — Yenus    passes 
through  the  same  changes  of  phase  as  Mercury.     But  its  ap- 
parent diameter,  when  the  crescent  phase  is  narrowest,  is  more 
than  6  times  as  great  as  when  at  full.     For  its  distance  from 
us,  in  the  former  case,  is  95,000,000  —  69,000,000  =  26,000,000 
miles ;    and   in   the  latter,   it    is   95,000,000  +  69,000,000  = 
164,000,000  miles,  a  distance  more  than  six  times  as  great  as 
the  other. 

Yenus  is  the  brightest  of  the  planets,  and  has  been  known 
from  ancient  times  as  the  morning  and  evening  star,  according 
as  it  is  west  of  the  sun,  or  east  of  it. 

The  place  of  greatest  brightness  for  Yenus  is  when  about  40° 
from  the  sun,  between  the  point  of  greatest  elongation  and  the 
inferior  conjunction.  In  this  situation,  it  is  frequently  visible 
all  day. 

279.  Transits  of  Venus. — The  orbit  of  Yenus  is  inclined 
to  the  ecliptic  about  3J  degrees.     The  sun  passes  its  nodes  in 
June  and  December;    therefore,  the   transits  of  that  planet 
occur  in  those  months. 

Yenus  makes  13  revolutions  in  very  nearly  the  same  time  in 
which  the  earth  makes  8.  Hence,  a  transit  of  Yenus  at  either 
node  is  usually  preceded  or  followed  by  another  at  the  same 
node,  at  an  interval  of  8  years.  But  this  interval  can  not  be 
halved,  as  in  the  case  of  Mercury  (Art.  275),  to  find  the  time 
of  a  transit  at  the  other  node ;  because,  8  being  an  even,  and 
13  an  odd  number,  there  would,  in  4  revolutions  of  the  earth, 
be  6J  revolutions  of  Yenus,  which  would  bring  the  two  planets 
on  opposite  sides  of  the  sun. 

The  interval  of  235  years  is  much  more  exactly  measured  by 
382  revolutions  of  Yenus.  Therefore,  after  a  transit,  there  is 
almost  a  certainty  of  another  at  the  same  node  in  235  years. 
But,  for  the  same  reason  as  before,  the  middle  of  this  interval 
can  not  be  taken  as  the  date  of  a  transit  at  the  other  node. 


PAEALLAX  OF   THE   SUN".  147 

The  smaller  intervals  must  be  obtained  by  using  the  period  of 
227  years,  which  is  8  years  less  than  235  years. 

In  227  years,  there  are  369  revolutions  of  Yenus  within 
1J  days.  Hence,  transits  are  very  likely  to  occur  at  the  same 
node  at  intervals  of  227  years.  And  at  the  middle  of  this  in- 
terval, there  will  probably  be  a  transit  at  the  other  node,  since 
1.1 3 J  revolutions  of  the  earth,  and  184J  of  Yenus,  bring  both 
bodies  to  the  opposite  side  of  the  heavens.  This  interval  of 
1131  years  may  be  increased  or  diminished  by  8,  to  furnish  two 
other  intervals.  Hence,  the  ordinary  intervals  are  8,  105-1, 
11 3^,  and  121^  years,  as  may  be  seen  in  the  following  series  of 
transits  from  1518  to  2004: 

Interval. 

June  5th,  1518 

June2d,    1526 8    years. 

Dec.  4th,  1639 113£  " 

June  5th,  1761 121J  « 

June  3d,    1769 8  «~- 

Dec.  8th,  1878 105$  <; 

Dec.  6th,  1882 8  " 

June  7th,  2004 121J  " 

280.  Parallax  of  the  sun  ~by  a  transit  of  Venus. — The 
planet  Yenus  is  so  near  the  earth,  that  its  transit  across  the 
sun's  disk  is  peculiarly  favorable  for  obtaining  the  sun's  paral- 
lax.    Let  E  (Fig.  70)  be  the  earth,  Y  Yenus,  and  fde  the  disk 
of  the  sun.     Suppose  observers  stationed  at  A.  and  B,  the  ex- 
tremities of  that  diameter  which  is  perpendicular  to  the  orbit 
of  Yenus.     Each  one  sees  the  planet  describe  a  chord  across 
the  sun's  disk  from  east  to  west.     A  observes  it  to  come  on  at 
c,  and  leave  at  d  ;  while  to  the  view  of  B,  it  comes  on  at  e?,  and 
leaves  at  f.     And  when  it  appears  at  a  to  the  former,  it  is  seen 
at  I)  by  the  latter.     It  is  the  distance  between  the  two  projec- 
tions at  a  and  b  which  is  to  be  determined. 

281.  The  length  of  oh  in  miles. — Since  the  periodic  times 
of  the  earth  and  Yenus  are  known,  the  ratio  of  the  distances  oi 
E  and  Y  from  the  sun  are  also  known,  by  Kepler's  third  law. 
Hence,  by  subtraction,  the  ratio  of  the  lengths  of  the  triangles 


14:8  EXTERNAL   AND   INTERNAL   CONTACTS. 

YA  and  Y#  is  known.  These  triangles  may  be  regarded  as 
isosceles ;  therefore,  as  they  have  equal  angles  at  Y,  they  are 
similar.  Hence,  YA  :  Y# : :  AB  :  ab.  Thus,  from  the  known 
ratio  of  YA  to  Ya,  and  the  length  of  AB,  we  have  the  length 
of  ab  in  miles. 

Fig.  70. 
V  B 


JT. 


282.  The  length  of  sib  in  seconds. — "We  next  wish  to  obtain 
the  angular  length  of  ab.     The  observers  carefully  mark  the 
moment  of  entering  on  the  disk,  and  the  moment  of  leaving  it. 
Thus,  the  length  of  time  occupied  by  the  transit,  as  seen  by 
each   observer,  is  carefully  obtained.     But  since  the  angular 
motion  per  hour,  both  of  the  planet  and  the  sun,  is  known,  the 
time  of  crossing  the  disk  can  be  changed  into  an  arc  ;  and  we 
thus  have  the  number  of  seconds  of  a  degree  in  the  chord  cd, 
and  also  the  number  in  ef,  and,  therefore,  in  their  halves,  ca 
and  eb.     Bnt  the  number  of  seconds  in  the  sun's  semi-diameter, 
<?S  or  <?S,  is  known.     Hence,  in  the  right-angled  triangles  cSa, 
eSb,  we  readily  find  the,  seconds  in  Sa  and  SZ>,  the  difference 
between  which  is  the  length  of  ab  in  seconds.     Thus,  we  find 
what  angle  is  subtended  by  a  line  of  given  length,  when  placed 
at  the  sun,  and  viewed  from  the  earth ;  or,  which  is  the  same 
thing,  placed  at  the  earth,  and  viewed  from  the  sun.     There- 
fore, we  know  what  angle  at  the  sun  is  subtended  by  the 
radius  of  the  earth ;  and  that  is  the  sun's  horizontal  parallax. 

283.  External  and  internal  contacts. — At  inferior  conjunc- 
tion, the  planet  Yenus  subtends  an  angle  of  more  than  T,  and 
therefore,  in  the  transit,  appears  like  a  small  black  circle,  whose 
diameter  is  ^y  of  the  sun's  diameter.   To  observe  the  beginning 
and  end  of  a  transit,  the  instant  of  external  contact  must  first 
be  noted,  and  afterward,  when  the  planet  has  come  wholly 
upon  the  disk,  the  time  of  internal  contact  also.     The  mean  of 
these  is  the  time  at  which  the  center  crosses  the  edge  of  the 


APPAEENT   MOTIONS   OF  MAES.  149 

disk.  The  duration  of  the  transit  is  the  interval  between  the 
moments  at  which  the  center  of  the  planet  enters  and  leaves 
the  disk. 

284.  Situation  of  the  observers. — The   observers  can  not 
probably  be  at  points  diametrically  opposite,  nor  can  they  re- 
main  stationary  during   the  transit,   on    account   of  diurnal 
motion  ;  therefore,  allowance  must  be  made  for  these  circum- 
stances.    In  order  that  several  independent  results   may  be 
obtained,  many  stations  are  chosen,  at  the  greatest  possible 
distance  from  each  other.     In  the  observations  on  the  transit  of 
1769,  one  of  a  large  number  of  stations  was  in  Lapland,  and 
another  on  one  of  the  Sandwich  Islands.     The  result  arrived  at 
was,  that  the  sun's  horizontal  parallax  is  8".5776,  which  has 
since  been  regarded  as  its  true  value. 

MAES. 

285.  Tabular  statements. — Mean   distance  from  the  sun, 
145,000,000  miles ;    periodic  time,  2  years ;    diameter,  4,500 
miles  ;  diurnal  rotation,  24.66  hours ;  specific  gravity,  3.9. 

286.  Situation  of  Mars  in  the  solar  system. — This  is  the 
most  remote  planet  of  the  first  group  described  in  Art.  256 — 
namely,  Mercury,  Yenus,  Earth,  Mars.     It  is  also  the  nearest 
to  the  earth  of  those  planets  which  are  called  superior. 

As  Mars  revolves  in  an  orbit  outside  of  the  earth's,  it  can 
come  into  opposition  to  the  sun,  as  well  as  into  conjunction 
with  it,  appearing  at  every  degree  of  elongation  from  0°  to 
180°. 

287.  Apparent  motions.-*-T1n.Q  real  motion  of  Mars  is  from 
west  to  east ;  and  during  most  of  the  year,  its  apparent  motion 
is  in  the  same  direction,  sometimes  accelerated,  and  sometimes 
retarded,  by  the  earth's  motion.     Near  opposition,  however, 
when  the  earth  overtakes  and  passes  by  Mars,  its  motion  ap- 
pears  retrograde.     Thus,  let  the  earth  make  one  revolution 
from  F  to  F  again  (Fig.  71),  while  Mars  describes  nearly  a  half 
revolution  from  G  to  ~N.    When  the  earth  is  at  F,  Mars  ap- 


150 


APPARENT   MOTIONS   OF   MARS. 


pears  in  the  direction  FGr ;  when  at  A,  Mars  at  H,  appears  in 
the  sky  at  O ;  when  the  earth  is  at  B,  Mars  at  I,  appears  at  P. 
Thus  far,  the  motion  has  been  in  advance,  though  becoming 
retarded  near  P.  But  as  the  earth  passes  from  B,  through  C, 
to  D,  Mars,  passing  over  the  shorter  arc  IKL,  appears  to  retro- 
grade from  P  to  Q ;  after  which  it  again  advances,  appearing 
at  E  when  the  earth  is  at  E  and  in  the  direction  FN  when 
the  earth  is  at  F. 

Fig.  71. 


For  the  same  reason,  all  the  superior  planets  have  a  retro- 
grade motion  at  the  time  of  opposition. 

288.  Phases,  and  changes  of  apparent  size. — At  opposition, 
M  (Fig.  72),  and  at  conjunction,  M','  it  is  obvious  that  Mars 
appears  full,  since  we  look  in  the  same  direction  in  which  the 
sun  shines  upon  it.  In  other  positions,  the  angle  between  the 
sun's  rays  and  our  visual  line  is  acute,  and  the  phase  is  gibbous 
(Art.  170).  The  planet  is  so  near  us,  that  the  phase  differs 
perceptibly  from  the  full,  when  afcout  half-way  from  conjunc- 
tion to  opposition,  as  at  Q,  Q'. 

At  opposition,  Mars  is  nearer  to  us  than  at  conjunction  by 
the  diameter  of  the  earth's  orbit.  This  makes  its  mean  distance 
at  opposition  50,000,000  miles,  and  at  conjunction,  240,000,000. 
But  on  account  of  the  elliptical  form  of  both  orbits,  the 
least  distance  is  35,000,000  miles,  and  the  greatest,  255,000,000 
miles. 


APPEAEAKCE   OF  DISK. 


151 


289.  Orbit  and  equator  of  Mars. — The  orbit  of  Mars  is  in- 
clined to  the  ecliptic  nearly  2°,  and  has  an  eccentricity  equal 
to  TV 


In  its  diurnal  rotation,  it  considerably  resembles  the  earth, 
having  about  the  same  length  of  day,  and  its  equator  being  in- 
clined nearly  29°  to  its  orbit.  Hence,  the  seasons  vary  some- 
what more  than  those  on  the  earth. 

29O.  Appearance  of  disk. — Mars  is  remarkable  among  the 
planets  for  its  redness.  The  telescope  reveals  some  permanent 
inequalities  of  surface,  by  which  its  diurnal  rotation  has  been 
determined  more  satisfactorily  than  in  the  cases  of  Mercury 
and  Yenus.  And  there  are  other  appearances,  which  change 
as  the  relation  of  the  equator  to  the  sun  changes.  The  polar 
regions,  when  turned  away  from  the  sun,  exhibit  a  whiteness, 
which  is  supposed  to  be  the  effect  of  ice  and  snow ;  and  this 
whiteness  disappears  gr:idually,  when  the  pole  is  turned  again 
toward  the  sun. 


152  GHAEACTERISTICS   OF   PLANETOIDS. 


CHAPTER  XVI. 

THE  PLANETOIDS. — JUPITER. — SATURN. — URANUS. — NEPTUNE. 

291.  The  space  between  the  four  small  planets  and  the  four 
large  ones. — The  large  interval   between  Mars   and  Jupiter, 
which  seemed  to  break  the  continuity  of  the  series  of  planets, 
was  noticed  by  Kepler.     About  the  close  of  the  last  century, 
Bode,  of  Berlin,  showed  that  a  series  of  numbers,  following  a 
certain  law,  would  express  pretty  accurately  the  planetary  dis- 
tances from  the  sun,  if  only  the  vacancy  between  Mars  and 
Jupiter  were  supplied.     This  led  to  a  special  search  for  new 
planets,  which  was  presently  rewarded  by  the  discovery  of  sev- 
eral small  bodies,  which  have  been  called  asteroids,  planetoids, 
or  minor  planets. 

THE    PLANETOIDS. 

292.  Their  number,  and  the  time  of  their  discovery. — Four 
of  these  bodies  were  discovered  within  the  first  seven  years  of 
the  present  century — namely  :  Ceres,  Pallas,  Juno,  and  Vesta. 
Since  1845,  others  have  been  found  nearly  every  year,  till  their 
number  at  the  present  time  (1866)  is  between  80  and  90.     The 
whole  number  of  planetoids  may  be  regarded  as  indefinitely 
great. 

293.  Characteristics. — They   are    distinguished   from   the 
eight  planets  in  the  following  particulars  : 

1.  By  their  diminutive  size. — They  are  invisible  to  the 
naked  eye,  and  by  the  telescope  can  not  be  distinguished  from 
faint  fixed  stars,  except  by  their  motion.  They  are  generally 
-•-,00  small  to  show  a  sensible  disk,  and  hence  can  not  be  meas- 
ured with  any  certainty.  The  largest  of  them  is  believed  to  be 
only  about  200  miles  in  diameter.  And  it  is  estimated  by  the 
slight  disturbing  influence  which  they  exert,  that  their  entire 
mass  is  equal  only  to  a  small  fraction  of  the  earth. 


JUPITER'S  MAGNITUDE.  153 

2.  By  the  large  eccentricity  and  obliquity  of  their  orbits. — 
The  eccentricity  of  most  of  them  is  much  greater  than  that  of 
any  of  the  eight  planets. 

The  obliquity  of  the  orbit  of  Hebe  is  14°,  and  that  of  Pallas 
is  34°,  which  is  the  greatest  yet  discovered. 

3.  By  their  being  clustered  in  a  ring. — The  orbits  vary  con- 
siderably in  size,  and  therefore  the  periodic  times  are  various. 
But  as  they  are  generally  quite  eccentric,  every  planetoid  is 
nearer  the  sun  at  perihelion,  than  any  other  one  is  at  aphelion. 
The  orbits  are  therefore  all  linked  together,  and  pass  through 
each  other.     Thus,  the  planetoids  are  to  be  regarded  as  moving 
among  each  other  about  the  sun,  within  the  limits  of  a  ring, 
whose  breadth,  in  the  direction  of  the  radius  vector,  is  more 
than  100,000,000  miles.     Flora,  which  moves  in  the  smallest 
orbit   yet   discovered,   performs    its   revolution    in   3^  years; 
Cybele,  the  most  remote,  in  6J  years.     Their  mean  periodic 
time  is  4^  years ;  and  their  mean  distance  from  the  sun  is 
254,000,000  miles. 

294.  Modes  of  designating  them. — Feminine  mythological 
names  have  been  applied  to  all  the  planetoids  which  have  yet 
been  discovered.     But  the  more  convenient  method,  and  the 
one  most  used,  is  to  express  each  planetoid  by  a  number,  show- 
ing its  place  in  the  order  of  discovery,  this  number  being  in- 
closed in  a  circle,  which  indicates  a  disk.     Thus,  Ceres  is  Q  ; 
Thetis,  @ ;  Pandora,  @ ;  etc.     See  Table  III.,  at  the  end. 

JUPITER. 

295.  Tabular  statements. — Mean   distance   from  the  sun, 
496,000,000  miles ;  periodic  time,  12  years ;  diameter,  89,000 
miles;  diurnal  rotation,  9.92  hours;  specific  gravity,  1.3. 

296.  Jupiter's  magnitude  and  place  in  the  solar  system. — • 
Jupiter  is  the  nearest  of  the  large  planets  outside  of  the  planet- 
oids, and  his  orbit  is  more  than  300,000,000  miles  beyond  the 
ring  which  includes  them.     On  account  of  its  great  distance 
from  the  sun,  compared  with  the  earth's,  Jupiter  presents  to  us 
no  visible  change  of  phase,  appearing  always  full.     Its  disk,  as 


154  THE  BELTS   OF  JUPITER. 

presented  to  us,  is  almost  the  same  as  if  we  were  at  the  sun. 
The  same  is,  of  course,  true  of  all  the  planets  still  more  re- 
mote. 

Jupiter  greatly  surpasses  all  the  other  planets  in  magnitude. 
In  volume,  it  is  about  1 J  times  the  sum  of  all  the  others,  and  in 
mass,  more  than  2^  times  their  united  mass. 

297.  Its  spheroidal  form. — Though  the  diameter  of  Jupiter 
is  11  times  that  of  the  earth,  yet  it  rotates  on  its  axis  in  less 
than  10  hours ;  so  that  the  equatorial  velocity  is  about  27  times 
as  great  as  the  earth's.     This  rapidity  of  rotation  produces  a 
sensible  oblateness  of  the  planet.     Its  ellipticity  is  ^\ ;  and  so 
considerable  a  deviation  from  the  spherical  form  is  perceptible 
to  the  eye  without  measurement. 

298.  The  belts  of  Jupiter. — This  name  is  given  to  bands  or 
stripes  of  darker  shade  than  the  rest  of  the  disk,  stretching 
across  it  in  the  direction  of  its  rotation  (Fig.  4,  Fr.)     They  vary 
from  time  to  time  in  number  and  in  breadth,  often  covering  a 
large  part  of  the  surface.     A  belt  usually  appears  of  uniform 
breadth  entirely  across,  but  not  always  ;  its  edge  is  occasionally 
broken,  and  sometimes  it  is  much  wider  on  one  part  of  the 
disk  than  on  the  other,  the  change  of  breadth  being  commonly 
quite  abrupt,  and  thereby  revealing  the  rotation  of  the  planet. 
There  are,  ordinarilyj  two  conspicuous  belts,  lying  near  the 
equator,  one  north,  and  the  other  south  of  it. 

299.  Supposed  cause  of  the  belts. — The  belts  are  considered 
as  affording  proof  that  Jupiter  is  surrounded  by  an  atmosphere, 
in  which  clouds  are  floating.     As  a  consequence  of  the  exceed- 
ingly rapid  rotation  of  the  planet,  there  would  be  very  power- 
ful currents,  analogous  to  the  trade-winds  of  the  earth ;    and 
the  clouds  would  be  thrown  into  the  form  and  arrangement  of 
zones  parallel  to  the  equator.     The  clouds  would  reflect  the 
sun's  light  to  us  more  strongly  than  the  atmosphere  ;  and  the 
dark  belts,  therefore,  are  the  unclouded  portions,  through  which 
we  look  on  the  body  of  the  planet. 

300.  Orbit  and  equator  of  Jupiter. — The  orbit  of  Jupiter 


SATELLITES  OF  JUPITER.  155 

is  nearly  coincident  with  the  plane  of  the  ecliptic,  its  inclina- 
tion being  only  1°  1 9'.  Its  eccentricity  is  ^V?  which  is  three 
times  as  great  as  that  of  the  earth's  orbit. 

The  equator  of  Jupiter  is  inclined  a  little  more  than  3°  to 
its  orbit.  There  is,  therefore,  no  perceptible  change  of  seasons 
on  that  planet. 

301.  Satellites  of  Jupiter. — These  are  four  in  number,  re- 
volving in   orbits  very  nearly  circular,  and  in  planes  which 
make  small  angles,  both  with  the  orbit  and  the  ecliptic.     They 
are  called  the  first,  second,  third,  and  fourth,  reckoning  out- 
ward from  the  planet. 

0 O2.  Their  revolutions. — On  account  of  the  position  of  the 
orbits,  we  see  the  satellites  passing  back  and  forth  across  the 
place  of  Jupiter,  nearly  in  straight  lines  (Fig.  4-,  Fr.)     From 
their  greatest  elongation  west  of  Jupiter,  they  advance  to  the 
greatest  elongation  on  the  east,  passing  behind  the  planet  on 
their  way.    Then,  after  remaining  stationary  a  short  time,  they 
retrograde  to  the  west  side,  passing  between  us  and  the  planet. 
These  movements  prove  that  they  revolve  from  west  to  east,  as 
all  the   primary  planets  do.     At  the  greatest  elongation  on 
the  east  side,  they  are,  for  a  little  while,  stationary,  because 
coming  toward  us ;  and  on  the  west  side  also,  because  going 
from  us. 

303.  Their  size,  distance,  and  periods. — Jupiter's  satellites 
are  all  somewhat  larger  than  the  earth's  moon,  the  smallest  ex- 
ceeding it  by  only  30  miles  in  diameter,  the  largest  by  1,420 
miles.     To  us,  they  appear  as  stars  of  the  6th  or  7th  magni- 
tude ;  but  on  account  of  the  brightness  of  the  primary,  they 
can  very  rarely,  if  ever,  be  seen  by  the  naked  eye.     If  two  or 
three  satellites  happen  to  appear  very  near  together,  they  may 
possibly  be  seen  by  the  naked  eye,  when  they  of  course  seem 
to  be  one.     The  first  is  further  from  Jupiter  than  the  moon  is 
from  the  earth,  and  the  fourth  more  than   five  times  as  far. 
Their  periods  of  revolution  are  very  short,  compared  with  the 
moon's;  for,  on  account  of  the  strong  attraction  of  Jupiter, 
great  velocity  is  requisite  to  maintain  them  in  their  orbits. 


156 


ECLIPSES  AND   OCCULT  ATIONS. 


Satellites. 

Diameters. 

Distances. 

Sidereal  Revolutions. 

1 

2,440 

278,500 

Id.  18  h.  28m. 

2 

2,190 

443,300 

3  "  13  "  15  " 

3 

3,580 

707,000 

7  "    3  "  43  " 

4 

3,060 

1,243,500 

16  "  16  "  32  " 

304.  Their  con-figurations. — The  relative  positions  of  Jupi- 
ter and  the  four  satellites,  as  seen  from  the  earth,  are  inces- 
santly varying.     We  most  frequently  see  two  or  three  on  one 
side,  and  two  or  one  on  the  other;   rarely  all  on  one  side. 
Yery  often,  one  or  two   are   invisible,  being   either  behind 
Jupiter,  or  projected  on  it.     Sometimes,  three  are  thus  con- 
cealed, and  in  very  rare  instances,  all  four. 

305.  Eclipses  and  occultations  of  Jupiter  and  its  satel- 
lites.— The  great  dimensions  of  Jupiter  and  its  shadow,  and 
the  small  inclinations  between  the  ecliptic,  Jupiter's  orbit,  and 
those  of  its  satellites,  cause  very  frequent  eclipses  and  occulta- 
tions.    A  satellite  of  Jupiter  is  eclipsed  when  it  goes  through 
the  shadow  of  the  planet;    it   suffers   occultation  when  it  is 
hidden  from  our  view  by  passing  behind  the  planet.    The  first, 
second,  and  third  satellites  pass  through  both  eclipse  and  oc- 
cultation at  every  revolution,  and  the  fourth  rarely  escapes. 

Besides  these  two  classes  of  phenomena,  there  are  two  others, 
— namely,  the  eclipse  of  Jupiter,  when  its  satellite  casts  a 
shadow  upon  it ;  and  an  occultation  of  Jupiter,  when  a  satel- 
lite passes  between  it  and  the  earth.  The  eclipse  is  a  small 
black  spot  passing  over  the  disk.  The  occultation  is  scarcely 
perceptible,  because  the  planet  and  satellite  are  of  about  equal 
brightness.  On  a  belt,  the  satellite  may  appear  brighter, 
and  between  two  belts  it  may  appear  less  bright  than  the 
primary. 

306.  Order  of  eclipses  and  occultations. — When  Jupiter  is 
east  of  opposition,  the  eclipse  always  precedes  the  occultation  ; 
when  west  of  opposition,  the  occultation  precedes  the  eclipse. 
For,  let  S  (Fig.  73)  be  the  sun ;  A,  B,  C,  several  positions  of 
the  earth  ;  J,  Jupiter ;  and  EHK,  the  orbit  of  a  satellite.     The 


ECLIPSES  AND   OCCULTATIONS. 


157 


bodies  are  supposed  to  revolve  in  the  order  of  the  letters.  If 
the  earth  is  at  A,  SA  produced  marks  the  place  of  opposition, 
and  Jupiter  is  east  of  that  place.  The  satellite  enters  the 
shadow  at  E,  emerges  at  F,  and  then  passes  behind  the  planet 
at  G,  and  reappears  at  H.  In  this  case,  the  eclipse  is  past  be- 
fore the  occultation  begins.  In  the  same  manner,  the  eclipse 
of  Jupiter  begins  when  the  satellite  is  at  K,  and  ends  when  at 
L ;  and  the  occultation  follows  it,  while  the  satellite  moves 
from  M  to  JST.  If  the  earth  were  at  C,  Jupiter  would  be  west 
of  opposition — that  is,  west  of  SO  produced.  And  it  is  obvious 
that  the  satellite  would  go  behind  the  planet  before  entering 
the  shadow,  and  also  would  appear  between  us  and  the  planet 
before  casting  a  shadow  on  it. 

Fig.  73. 


The  earth  is  not,  in  general,  so  situated  that  one  phenom- 
enon is  closed  before  the  next  begins ;  and  it  is  never  true  of 
the  first  satellite.  The  case  is  represented  by  the  orbit  ehkn. 
The  eclipse  begins  at  e,  and  the  occultation  ends  at  h  ;  but  the 
end  of  the  eclipse  and  the  beginning  of  the  occultation  are  not 
seen.  In  the  same  manner,  the  eclipse  of  Jupiter  begins  at  &, 
and  the  occultation  ends  at  n.  During  a  part  of  the  interven- 
ing time,  the  shadow  and  the  body  of  the  satellite  are  both 
seen,  projected  at  different  places  on  the  primary. 

At  the  time  of  opposition,  the  earth  being  at  B,  the  eclipse 


158  SATURN'S  DISK. 

of  a  satellite  obviously  occurs  entirely  within  its  occultation, 
and  the  occultation  of  Jupiter  entirely  within  its  eclipse. 

It  is  found  that  there  exists  such  a  relation  between  the 
mean  motions  of  the  three  first  satellites,  that  they  can  never 
all  be  eclipsed  at  the  same  time. 

307.  The  velocity  of  light  discovered  by  the  eclipses  of 
Jupiter's  satellites. — In   1675,  it  was  discovered  by  Roemer 
that  eclipses  occurred  earlier  than  the  calculated  time,  when 
the  earth  is  in  that  part  of  its  orbit  which  is  near  to  Jupiter, 
and  later,  when  in  the  remote  part.     The  eclipses  of  any  one 
satellite  are  so  frequent,  that  the  mean  interval  between  them 
is  obtained  with  great  accuracy ;  and  by  this  mean  interval, 
the  times  of  future  eclipses  could  be  calculated.     But  it  was 
perceived  that  while  the  earth  moves  from  the  remote  side  to 
the  nearer  side  of  its  orbit,  the  real  intervals  are  shorter  than 
the  mean,  so  that,  at  the  nearest  point,  an  eclipse  occurs  about 
Sm.  13^s.  too  soon.     Again,  as  the  earth  goes  to  the  side  of  its 
orbit  furthest  from.  Jupiter,  the  real  intervals  are  all  greater 
than  the  mean ;   and  at  the  most  distant  point,  an  eclipse  is 
later  than  the  calculated  time  by  8m.  13^s.    Eoemer  attributed 
this  periodical  error  of  time  to  the  progress  of  light,  and  in- 
ferred that  light  requires  16m.  27s.  to  cross  the  earth's  orbit. 
This  makes  the  velocity  of  light  near  200,000  miles  per  sec- 
ond ;  which  seemed  at  first  quite  incredible,  and  was  received 
with  distrust.     But  its  correctness  was  soon  established  by  the 
discovery  of  the  aberration  of  the  stars,  which  gives  about  the 
same  result  (Art.  146). 

SATURN. 

308.  Tabular    numbers — Mean    distance   from    the    sun, 
909,000,000  miles ;   periodic  time,  29  years ;  diameter,  79,000 
miles  ;  diurnal  rotation,  10.48  hours  ;  specific  gravity,  0.7. 

0 O9.  Satur-tJs  dit-k. — Saturn  is  the  second  planet  in  size; 
and  being  the  second  in  order  beyond  the  planetoids,  is  not  too 
far  from  the  earth  to  present  a  large  disk.     Its  form  is  seen  to 
be  elliptical,  and  it  is  faintly  striped  with  belts  in  the  direction 


SATURN'S  RINGS.  159 

of  the  major  axis.  Both  these  appearances  are  explained  by 
the  rapid  rotation  of  the  planet  on  its  axis,  as  in  the  case  of 
Jupiter.  Its  ellipticity  is  T^. 

310.  Saturn's  rings. — The  distinguishing  feature  of  this 
planet  is  the  system  of  broad  thin  rings  which  surround  it. 
They  lie  in  a  plane  inclined  about  28°  to  the  ecliptic,  and 
therefore  generally  present   an   elliptical  appearance    to   the 
earth  (Fig.  3,  Fr.)     The  ring,  as  usually  seen,  consists  of  two 
rings,  the  inner  of  which  is  the  widest.     The  inner  edge  is 
20,000  miles  from  the  surface  of  the  planet ;  and  the  diameter 
from  outside  to  outside  is  176,000  miles.     The  line  in  which 
the  plane  of  the  ring  intersects  the  plane  of  Saturn's  orbit  is 
called  the  line  of  the  nodes. 

"Within  the  double  ring  already  described,  there  is  a  much 
fainter  one,  which  can  not  be  seen  with  ordinary  telescopes. 
By  careful  observations,  it  is  also  perceived  that  there  are  sev- 
eral concentric  divisions  of  the  rings,  which  vary  their  number 
and  position  from  time  to  time.  These  fainter  divisions  are 
invisible,  except  at  the  ends  of  the  ellipse.  The  rings  lie  in 
one  plane,  and  are  exceedingly  thin.  The  latest  measurements 
make  their  thickness  less  than  40  miles.  A  circle  of  common 
writing  paper,  one  foot  in  diameter,  would  be  too  thick  to  rep- 
resent it  correctly.  But  the  thickness  appears  not  to  be  uni- 
form ;  for  in  the  edge  view,  it  often  presents  the  aspect  of  a 
broken  line,  as  though  some  parts  were  thick  enough  to  be  seen, 
and  others  not.  There  seems  to  be  evidence  that  the  rings 
consist  either  of  liquid  matter,  or  else  of  solid  matter  in  a  dis- 
integrated condition. 

311.  Rotation  of  the  rings. — Such  rings  of  matter  around 
Saturn  could  no  more  be  sustained  without  rotation,  than  the 
moon  could  remain  at  its  distance  from  the  earth  without  re- 
volving about  it.     They  are  found  to  rotate  in  their  own  plane 
within  the  short  period  of  10 1  hours,  the  same  as  the  period  of 
the  planet  itself.     The  outer  edge  of  the  ring  must,  therefore, 
have  a  velocity  of  14  or  15  miles  per  second. 

3 1 2.  The  plane  of  the  rings  always  parallel  to  itself. — 


160 

During  the  revolution  of  Saturn  around  the  sun,  occupying 
about  29  years,  the  rings  maintain  everywhere  the  same  posi- 
tion in  relation  to  the  plane  of  Saturn's  orbit  as  represented  in 
Fig.  74,  in  which  a~b  is  the  earth's,  and  ACEG  Saturn's  orbit, 
seen  obliquely.  While  the  planet  passes  through  the  half  revo- 
lution ACE,  the  north  side  of  the  rings  is  seen  by  an  observer 
on  the  earth  as  an  ellipse,  more  or  less  eccentric ;  but  during 
the  other  half,  EGA,  the  south  side  is  in  view.  Each  of  these 
periods  occupies  near  15  years.  When  Saturn  is  near  A  and 
E,  the  line  of  nodes  passes  across  the  earth's  orbit,  and  the 
edge  of  the  rings  is  therefore  directed  toward  the  sun  and 
earth ;  and  at  those  times  it  fills  too  small  an  angle  to  be  seen, 
except  by  the  best  instruments. 

Fig.  74. 


313.  Passage  of  the  plane  of  the  rings  across  the  earth's 
orbit. — The  motion  of  Saturn  is  so  slow,  that  it  requires  almost 
a  year  for  the  plane  of  its  rings  to  pass  by  the  whole  diameter 
of  the  earth's  orbit.  Let  DF  (Fig.  To)  be  the  earth's  orbit,  and 
AC  a  portion  of  Saturn's.  Suppose  these  orbits  to  lie  in  the 
plane  of  the  paper,  and  the  plane  of  the  rings  to  be  inclined 
about  28°  to  the  paper,  making  the  common  section  of  the  two 
planes  in  the  lines  AD,  BG,  etc.  Saturn  is  9.54  times  as  far 
from  the  sun  as  the  earth  is.  Therefore,  SA  :  SD  :  :  9.54  :  1 
:  :  rad.  :  sin  SAD ;  .-.  SAD,  or  its  equal,  ASB  =  6°  V ;  /.  ASC 
=  12°  2'.  Knowing  Saturn's  periodic  time,  we  readily  find 
that  it  will  describe  12°  2'  in  359J  days,  near  six  days  less  than 
a  year.  Hence,  while  Saturn  passes  from  A  to  C,  the  earth 
will  pass  very  nearly  around  its  orbit,  DEFG.  But  the  earth 


DISAPPEARANCE   OF  THE   RINGS. 


161 


may  be  at  any  point  of  its  orbit  when  the  planet  reaches  A. 
The  disappearances  of  the  rings  will  vary  according  to  the 
positions  of  the  earth. 

Fig.  75. 


3 1  4.  C'  r^nmstances  of  the  disappearances. — There  are  three 
ways  in  which  the  rings  may  fail  to  be  visible  during  the 
period  in  which  the  line  of  their  nodes  is  crossing  the  earth's 
orbit. 

1.  The  ring  may  present  its  edge  exactly  to  the  earth,  when, 
in  common  telescopes,  it  subtends  too  small  an  angle  to  be 
seen. 

2.  It  may  present  its  edge  exactly  to  the  sun,  so  that  neither 
side  of  the  ring  is  enlightened. 

3.  Itc  plane  may  be  directed  between  the  earth  and  sun,  when 
the  dark  side  is  toward  us. 

The  disappearance  by  either  of  the  two  first  causes  may  be 
considered  as-tmly  momentary;  for  the  line  of  nodes  passes 
the  breadth  of  the  sun  in  less  than  2  days,  and  of  the  earth  in 
about  20  minutes.  But  the  third  cause  may  conceal  the  ring 
from  our  view  for  weeks  or  months.  This  prolonged  disappear- 
ance may  occur  either  once  or  twice,  or  possibly  not  at  all,  while 
the  line  of  nodes  is  passing  the  breadth  of  the  earth's  orbit. 

315.  One  disappearance. — If  the  earth  is  at  F  when  the 
planet  reaches  A,  then  the  earth  wfll  go  from  F  nearly  to  D, 
while  the  nodal  line  advances  from  AD  to  BS,  and  the  earth 
will  pass  the  line  between  G  and  D,  as  at  K.  Up  to  that 
point,  the  luminous  side  is  presented  toward  the  earth ;  but 
from  K  to  a  point  near  D,  the  plane  of  the  rings  falls  between 
the  earth  and  sun,  and  the  rings  are  invisible,  and  continue  so 

11 


162  DISAPPEAKANCE   OF  THE   KINGS. 

about  two  months.  When  the  nodal  line  has  passed  the  sun, 
the  luminous  side  of  the  rings  is  again  toward  the  earth  ;  and 
before  the  earth  completes  the  half  orbit  DEF,  the  nodal  line 
will  pass  off'  at  F. 

316.  Two  disappearances. — If  the  earth  has  advanced  some 
distance  on  the  quadrant  FG — for  example,  to  the  middle  L — 
when  the  nodal  line  touches  D,  then  the  earth  passes  the  line 
between  K  and  D,  and  the  dark  side  is  toward  us.     The  line 
passes  the  sun  when  the  earth  is  near  the  middle  of  DE,  after 
which,  the  rings  are  seen.     But  before  the  nodal  line  reaches 
CF,  the  earth  will  overtake  it,  and  be  on  the  dark  side  again. 
Between  F  and  L,  the  earth  once  more  crosses  the  line,  and  the 
rings  present  to  us  their  bright  side.     In  this  case,  the  rings 
disappear  twice  during  the  nodal  year. 

These  two  periods  of  disappearance  may  be  so  prolonged  as 
to  unite  in  one  of  about  eight  months  in  length.  This  happens 
when  the  earth  is  two  or  three  days  past  G,  at  the  time  when 
the  nodal  line  touches  D.  Then,  before  reaching  D,  the  earth 
passes  to  the  dark  side  of  the  rings,  and  continues  on  that  side 
till  both  the  earth  and  the  nodal  line  pass  E  together.  As 
soon  as  that  point  has  been  passed,  the  line  is  again  between 
the  sun  and  earth,  and  continues  so  until  it  is  recrossed  by  the 
earth  on  the  quadrant  FG. 

317.  No  disappearance. — It  is  possible  that  no  disappear- 
ance, which  has  continuance,  should  happen  during  the  nodal 
year.     Suppose  the  earth  two  or  three  days  past  E,  when  the 
line  of  nodes  reaches  D.     Then,  while  the  line  moves  from  AD 
to  BS,  the  earth  will  advance  to  G,  all  the  time  on  the  lumi- 
nous side  of  the  rings ;  the  earth  and  sun  will  both  be  in  the 
line  BSG  at  once,  the  planet  being  in  conjunction  ;  and  after 
the  earth  has  passed  G  toward  D,  the  bright  side  of  the  rings 
is  in  view,  as  before,  and  will  continue  so.     Thus,  there  is  only 
a  momentary  disappearance,  and  that,  when  the  planet  and 
rings  are  lost  in  the  blaze  of  the  sun's  light. 

In  general,  there  are  two  periods  of  disappearance  within 
the  nodal  year,  arising  from  the  third  cause,  each  beginning 
and  ending  with  a  disappearance  from  the  first  or  second  cause. 


DISCOVERY   AND    PLACE    OF    URANUS.  163 

318.  Phenomena   of  the   rings   at   the  planet. — On   that 
hemisphere  of  the  planet  to  which  the  luminous  side  of  the 
rings  is  presented,  there  is  the  appearance  of  splendid  arches 
spanning  the  sky,  having  a  breadth  and  elevation  according  to 
the  latitude  of  the  place.    At  latitude  30°,  the  breadth  is  about 
18°,  and  the  elevation  of  the  lower  edge  on  the  meridian  about 
22°.     Near  the  poles,  however,  it  is  below  the  horizon.     The 
luminous  side  is  presented  to  the  northern  hemisphere  near  1 5 
years,  and  then  the  same  length  of  time  to  the  southern  hemi- 
sphere, in  regular  alternation. 

A  part  of  the  rings  is  generally  eclipsed  by  the  shadow  of  the 
planet  falling  on  it. 

Also,  during  the  15  years  in  which  the  dark  side  of  the 
rings  is  turned  toward  a  hemisphere,  its  shadow  is  cast  across 
a  zone  of  it,  which  causes  an  eclipse  of  the  sun.  And  at  a 
given  place,  a  total  solar  eclipse  may  continue  from  day  to  day, 
without  interruption,  for  several  years. 

319.  Satellites  of  Saturn. — Saturn   is  attended-  by  eight 
satellites.     Their  periods  of  revolution  vary  from  less  than  one 
day  to  79   days.      Their  diameters  vary  from  500  to  2,850 
miles ;   but  on  account  of  their  immense  distance  from  the 
earth,  they  are  seen  only  with  the  best  instruments.     They  are 
all  external  to  the  rings,  at  distances  from  the  planet,  varying 
from  118,000  to  2,30u,000  miles.     Their  orbits  are  nearly  in 
the  plane  of  the  rings,  and  make  an  angle  of  about  28°  with 
the  orbit  of  the  planet.     Hence,  they  are  not  very  liable  to  be 
eclipsed.     The  principal  time  for  eclipses  is  that  at  which  the 
rings  disappear  ;  for  then  the  sun  is  nearly  in  the  plane  of  their 
orbits,  as  well  as  of  the  rings. 

URANUS.  * 

3 2O.  Tabular  statements. — Mean   distance  from   the  sun, 
1,828,000,000  miles  ;  periodic  time,  84  years  ;  diameter,  35,000 
miles  ;  specific  gravity,  0.8. 

321.  Discovery,  and  place  in  the  system. — Uranus  was  un- 
known to  the  ancient  astronomers ;   and  to  them,  therefore, 


164  DISCOVEEY  OF  NEPTUNE. 

Saturn's  orbit  was  the  boundary  of  the  solar  system.  Uranus 
was  discovered  by  Sir  William  Herschel,  in  1Y81,  and  has 
made  but  little  more  than  one  revolution  since  that  time.  It 
was,  however,  repeatedly  seen  by  earlier  astronomers,  and  re- 
corded in  their  catalogues  as  a  fixed  star.  By  this  discovery, 
the  diameter  of  the  known  solar  system  was  doubled. 

Uranus  is  the  third  of  the  four  great  planets,  both  in  size 
and  in  order  of  distance.  But  its  distance  from  us  is  so  im- 
mense that  it  appears  only  as  a  faint  star,  and  presents  no  in- 
equalities by  which  its  diurnal  motion  can  be  discovered.  Its 
orbit  is  very  nearly  circular,  and  is  inclined  less  than  a  degree 
to  the  ecliptic. 

322.  The  satellites  of  Uranus. — Sir  William  Herschel  an- 
nounced the  discovery  of  six  satellites  belonging  to  Uranus. 
But  only  four  have  been  identified  by  later  astronomers.     The 
remarkable  facts  relating  to  these  satellites  are,  that  their  orbits 
are  nearly  at  right  angles  to  the  plane  of  the  ecliptic,  and  that 
in  the  orbits,  the  motions  of  the  satellites  are  retrograde — that 
is,  from  east  to  west.     Their  periods  of  revolution  vary  from 
2£   days  to  13J  days,  and  their  distances  from   120,000   to 
380,000  miles. 

NEPTUNE. 

323.  Tabular  statements. — Mean  distance  from  the   sun, 
2,862,000,000    miles;    periodic    time,    164   years;    diameter, 
31,000  miles  ;  specific  gravity,  1.5. 

324.  Discovery. — Neptune  was  discovered  in  1846.     The 
circumstances  which  led  to  the  discovery  were  briefly  as  fol- 
lows.   After  the  orbit  of  Uranus  had  been  carefully  computed, 
and  corrections  made  for  the  disturbing  influence  of  Jupiter 
and  Saturn,  the  planet  was  found  to  depart  from  the  calculated 
path  in  a  manner  not  to  be  accounted  for,  except  by  suppos- 
ing some  other  disturbing  force.     It  was  for  some  time  sus- 
pected that  there  must  be  a  planet  superior  to  Uranus,  whose 
attraction  caused  the   change  of  its  orbit.     At   length,  two 
mathematicians,  Le  Vender,  of  France,  and  Adams,  of  England, 


ELEMENTS   OF   DEBITS.  165 

each  without  any  knowledge  of  what  the  other  was  attempting, 
engaged  in  the  arduous  labor  of  calculating  what  must  be  the 
elements  of  a  planet  which  should  produce  the  given  disturb- 
ance of  the  motions  of  Uranus.  They  reached  results  which 
agreed  remarkably  with  each  other.  Le  Verrier  communicated 
to  Galle,  of  the  Berlin  observatory,  the  place  in  the  sky  in 
which  the  disturbing  body  should  be  situated ;  and  in  the 
evening  of  the  same  day,  Galle  found  it  within  a  degree  of  the 
predicted  longitude. 

The  planet  thus  discovered  explains  fully  the  disturbances  in 
the  motions  of  Uranus. 

It  soon  appeared  that  Neptune  had  repeatedly  been  entered 
in  catalogues  as  a  fixed  star.  The  earliest  of  these  records,  in 
1795,  afforded  material  aid  at  once  in  determining  its  mean 
distance  and  its  periodic  time. 

Neptune  is  attended  by  one  satellite,  which  was  also  dis- 
covered in  1846.  It  is  nearly  as  far  from  the  primary  as  the 
moon  is  from  the  earth,  and  revolves  in  5d.  21h. 


CHAPTER  XVII. 

ELEMENTS  OF  A  PLANETARY  ORBIT. — QUANTITY  OF  MATTER 
IN  THE  SUN  AND  PLANETS. — PLANETARY  PERTURBATIONS. — 
RELATIONS  OF  PLANETARY  MOTIONS. 

325.  Elements  of  an  orbit. — These  are  the  quantities  which 
must  be  known,  in  order  to  calculate  the  place  of  a  planet  at  a 
given  time.  They  are  seven  in  number. 

1.  The  periodic  time. 

2.  The  mean  distance  from  the  s*un,  or  the  semi-major  axis 
of  the  orbit. 

3.  The  longitude  of  the  ascending  node. 

4.  The  inclination  of  the  plane  of  the  orbit  to  that  of  the 
ecliptic. 

5.  The  eccentricity  of  the  orbit. 

6.  The  longitude  of  the  perihelion. 


166  PERIODIC  TIME. 

7.  The  place  of  the  planet  in  its  orhit  at  a  given  epoch. 

Two  of  these,  3d  and  4th,  determine  the  position  of  the  plane 
in  which  the  orbit  lies  ;  the  second  fixes  the  size  of  the  orbit ; 
the  5th,  its  form ;  the  6th,  the  relation  of  the  form  to  the  plane 
of  the  ecliptic ;  the  1st  and  7th,  the  circumstances  of  the 
planet's  motion  in  the  orbit. 

The  orbit  of  a  planet  can  not  be  determined  as  the  moon's 
orbit  is  (Chap.  X.),  or  the  sun's  apparent  orbit  (Chap.  IV.),  be- 
cause it  is  not  the  earth,  but  the  sun,  which  occupies  the  center 
of  the  planetary  revolutions. 

326.  Geocentric  and  heliocentric  place  of  a  planet. — The 
point  in  the  celestial  sphere  which  a  planet  occupies,  as  seen 
from  the  earth,  is  called  its  geocentric  place ;  its  place  as  seen 
from  the  sun  is  called  its  heliocentric  place.     It  has  already 
been  noticed  that  the  planets,  as  seen  from  the  earth,  have  a 
retrograde  motion  during  a  part  of  every  synodical  revolution. 
This  is  a  parallactic  effect  of  the  observer's  motion,  and  would 
not  exist  if  he  were  stationed  at  the  sun.   The  place  of  a  planet, 
as  seen  from  the  earth   and  the  sun,  can  never  agree,  except 
when  the  sun  and  earth  are  on  the  same  side  of  the  planet,  and 
in  the  same  straight  line  with  it.     But  after  the  relations  of  the 
earth  to  the  planet  and  to  the  sun  are  obtained,  there  is  no  dif- 
ficulty in  calculating  the  heliocentric  place  of  the  planet. 

327.  First  dement — the  periodic  time. — This  is  found  by 
observing  the  time  that  intervenes  between  the  two  successive 
returns  of  the  planet  to  the  same  node. 

It  may  be  known  when  a  planet  is  at  a  node,  because  then 
its  latitude  is  nothing.  If,  from  a  series  of  observations  on  the 
light  ascension  and  declination  of  a  planet,  the  latitudes  are 
computed,  and  one  of  them  is  zero,  then  the  exact  time  of  pass- 
ing the  node  is  obtained.  *  But  if,  as  is  usually  the  case,  the 
two  least  latitudes  are,  one  north,  and  the  other  south,  the  time 
of  passing  the  node  between  them  is  readily  found  by  a  propor- 
tion. Similar  observations  are  made  when  the  planet  again 
arrives  at  the  same  node,  and  thus  the  periodic  time  becomes 
known. 

It  is  discovered  that  a  minute  correction  of  the  periodic  time, 


DISTANCE   FROM   THE   SUN.  167 

thus  derived,  must  be  applied  for  tlie  retrograde  motion  of  the 
node.  The  periodic  time  of  a  planet  may  also  be  derived  from 
the  observed  length  of  its  synodic  revolution — that  is,  the  inter- 
val between  two  successive  oppositions,  or  two  conjunctions  of 
the  same  kind.  The  computation  is  similar  to  that  employed 
in  finding  the  sidereal  period  of  the  moon  from  its  synodical 
period  (Art.  158). 

In  both  the  above  methods,  great  advantage,  in  point  of  ac- 
curacy, is  gained,  if  two  very  distant  epochs  can  be  brought 
into  comparison,  such  as  two  distant  passages  of  the  node,  or  of 
opposition.  For  example,  a  transit  of  Mercury  occurs  at  in- 
ferior conjunction.  Divide  the  interval  between  two  observed 
transits,  several  years  apart,  by  the  number  of  synodical  revo- 
lutions of  Mercury  which  intervene,  and  its  mean  synodical 
period  is  very  accurately  obtained. 

328.  Second  element — the  distance  from  the  sun. — The  dis- 
tance of  an  inferior  planet  from  the  sun  is  found  as  follows. 
Let  S  (Fig.  76)  be  the  sun.  E  the  earth,  and  C 
the  planet.  Measure  the  greatest  elongation, 
SEC ;  then,  in  the  right-angled  triangle,  rad  : 
sin  SEC  :  :  SE  :  SO.  If  the  orbit  is  elliptical, 
the  value  of  SO,  as  obtained  at  different  times, 
will  be  different ;  and  a  great  number  of  such 
observations  should  be  made,  in  order  to  obtain 
the  mean  distance. 

The  distance  of  a  superior  planet  may  be 
found  by  observations  on  its  retrograde  motion 
at  the  time  of  opposition.  For,  the  more  dis- 
tant the  planet,  the  less  will  the  earth's  motion 
throw  it,  apparently,  backward.  Let  S  (Fig.  77) 
be  the  sun,  E  the  earth,  and  M  a  superior  planet. 
Let  E  pass  over  E^  in  a  short  time,  as  one  day,  and  let  M  pass 
over  Mm  in  the  same  time.  As  the  periodic  times  of  E  and  M 
are  supposed  to  be  known,  the  angles  ESe  and  MSm  are 
known,  and,  therefore,  their  difference,  eSm.  Join  em,  and 
produce  it  to  X  in  SM  produced.  Draw  ey  parallel  to  SX ; 
the  angle  ~X.ey  is  the  retrogradation  during  the  day  in  which 
the  planets  describe  the  arcs  E^  and  Mm,  and  is  known  by  ob- 
servation. But  SXe  =  ~X.ey  •  and,  therefore,  in  the  triangle 


168 


LONGITUDE   OF  THE  NODE. 


eSX,  the  third  angle,  X«?S,  is  known.  Henc3,  in  the  triangle 
Sem  we  have  all  the  angles,  and  the  side  Se9  by  which  Swi  is 
easily  computed.  This  process  may  be  repeated  at  every  oppo- 
sition, and  thus  the  mean  distance  is  ultimately  obtained. 


Fig.  78. 


329.  Third  element— longitude  of  the  node.— Let  S  (Fig. 
78)  be  the  sun,  EFG  the  earth's  orbit,  OPQ  the  orbit  of  a 
planet,  CL  an  arc  in  the  plane  of 
the  ecliptic,  intersecting  the  orbit 
in  P.  SP  is,  therefore,  the  line  of 
the  nodes.  And  let  EA,  FA7,  and 
SA"  be  parallel  lines,  directed  to- 
ward the  vernal  equinox.  When 
the  earth  is  at  E,  suppose  the  plan- 
et is  at  the  node  P ;  then  E,  P,  and 
S  are  all  in  the  plane  of  the  eclip- 
tic, and  AEP  is  the  longitude  of  P, 
and  AES  that  of  the  sun.  These 
longitudes  being  obtained,  their  dif- 
ference is  SEP,  which  is,  therefore, 
known.  After  the  planet  has  per- 
formed a  revolution  to  the  same 
node  again,  suppose  the  earth  to  be 
at  F;  then  we  find,  as  before,  its 
longitude,  ATP,  that  of  the  sun,  c 
A'FS,  and  their  difference,  SFP. 
As  the  times  are  known  in  which  ° 

the  earth  is  at  E  and  at  F,  we  know  SE,  SF,  and  the  angle  ESF, 
and  can  compute  EF,  and  the  angles  SEF  and  SFE.  From 
the  several  angles  at  E  and  F,  thus  obtained,  we  derive  PEF 
and  PFE ;  and  these,  with  the  side  EF,  give  us  the  side  FP. 
Then,  in  the  triangle  SFP,  from  SF,  FP,  and  SFP,  compute 
the  angle  FS  P.  From  this,  subtract  A"SF  (the  supplement  of 
ATS),  and  there  remains  A"SP,  the  heliocentric  longitude  of 
the  node. 

It  is  by  processes  of  this  kind  that  the  slow  retrograde  mo- 


INCLINATION  TO   THE'  ECLIPTIC.  169 

tion  of  the  nodes  is  discovered  (Art.  327).     It  amounts  to  only 
a  few  minutes  in  a  century. 

33O.  Fourth  element  —  inclination  of  the  orbit  to  the  eclip- 
tic. —  Select  the  time  of  observation,  when  the  sun's  longitude, 
obtained  from  the  tables,  is  the  same  as  the  heliocentric  longi- 
tude of  the  node  ;  and  find  for  that  time  the  geocentric  longi- 
tude and  latitude  of  the  planet.  Let  E  (Fig.  79)  be  the  earth, 
S  the  sun,  P  the  planet,  ISTO  the  line  of  the  nodes  coinciding 
with  ES  ;  and  let  EA  and  SA'  be  the  direction  of  the  vernal 
equinox.  Join  EP,  and,  with  it  as  a  radius,  describe  the  sur- 
face of  a  sphere,  cutting  the  plane  of  the  ecliptic  in  the  arc  BC. 
From  P  draw  the  arc  PQ,  perpendicular  to  BC.  AEO  is  the 
longitude  of  the  sun,  and  .A/SO,  its  equal,  is  the  heliocentric 
longitude  of  the  node  O.  AEQ  is  the  geocentric  longitude  of 
the  planet.  In  the  spherical  triangle  BPQ,  right-angled  at  Q, 
PQ  measures  the  given  latitude,  BQ  measures  the  difference 
between  AEQ  and  AES,  and  PBQ  is  the  inclination  to  be 
found.  Then,  rad  x  sin  BQ  =  tan  PQ  x  cot  PBQ  ; 


and  the  inclination  of  the  orbit  to  the  ecliptic  becomes  known. 

Fig.  79. 


331.  To  find  the  heliocentric  longitude  and  latitude  of  a 
planet.— Let  S  (Fig.  80)  be  the  sun,  E  the  earth,  EBC  its  orbit, 
P  the  planet,  EA,  SA'  the  direction  of  the  vernal  equinox. 
Let  PQ  be  drawn  perpendicular  to  the  plane  of  the  ecliptic. 
AEQ  is  the  geocentric  longitude  of  the  planet,  A'SQ  its  helio- 


1TO  ECCENTRICITY   OF  THE  ORBIT. 

centric  longitude.  Also,  PEQ  is  its  geocentric  latitude,  and 
PSQ  its  heliocentric  latitude.  SEP,  the  elongation  of  the 
planet  from  the  sun,  is  known  from  observation ;  SE,  the 
radius  vector  of  the  earth's  orbit,  and  SP,  that  of  the  planet's 
orbit,  are  also  known.  Therefore,  PE  may  be  computed. 
Knowing  PE,  and  PEQ  in  the  right-angled  triangle,  we  can 
compute  EQ.  Then,  in  the  triangle  QES,  EQ,  ES,  and  the 
angle  QES  (=  AES  -  AEQ)  being  known,  QSE  and  QS  are 
found.  From  QSE,  subtract  ESA'  (the  supplement  of  AES), 
and  A'SQ  is  obtained,  which  is  the  heliocentric  longitude  of  P. 
Again,  in  the  right-angled  triangle  PSQ,  having  SQ  and  SP, 
we  find  the  angle  PSQ,  the  heliocentric  latitude. 

Fig.  80. 


332.  Fifth  and  sixth  elements — eccentricity  of  the 
and  longitude  of  Ike pvi'ilwlioii. — A  focus  and  three  points  in 
the  curve  of  a  conic  section  being  given,  its  directrix  can  be 
determined,  and  the  curve  drawn.  (Coffin's  Con.  Sec.,  Prop. 
II.)  Thus,  let  SM,  SN",  and  SP  (Fig.  81)  be  three  radii  vec- 
tores  of  an  orbit,  determined  in  length  and  position  by  the  pro- 
cesses already  described.  If  MN  and  NP  are  joined,  the  tri- 
angles MISTS  and  NPS  are  known  in  all  respects.  Then,  if 
MN  be  produced,  so  that  NK  :  MR  :  :  NS  :  MS,  E  is  a  point 
of  the  directrix.  Another  point,  L,  is  fixed  in  a  similar  man- 
ner by  producing  PN.  The  directrix  being  thus  determined, 
draw  perpendiculars  to  it  from  S,  M,  1ST,  and  P.  The  axis  of 
the  orbit  is  on  KS  produced.  The  ratio,  SM  :  MG,  is  constant 
for  every  point  of  the  curve.  (Cof.  Con.  Sec.,  Pr.  II.) 


MASSES   OF   BODIES  fcOMPAKED. 
Fig.  81. 


171 


The  distance  of  the  focus  from  the  directrix  SK  is  found 
thus.  Draw  MD  perpendicular  to  SK.  LNS,  the  external 
angle  of  the  triangle  NSP  being  known,  subtract  MNS  from 
it,  and  we  have  LNR,  and  the  including  sides  LN",  KE,  to  find 
the  angle  R.  This,  with  the  side  ME,  in  the  right-angled  tri- 
angle MGR,  gives  us  GM,  and  the  angle  GMR.  Then, 
180°  -  (GME  +  EMS)  -  MSD,  from  which,  and  MS,  we 
compute  DS  ;  and  GM  +  DS  =  SK. 

To  find  the  perihelion,  divide  SK,  so  that  SA  :  AK  : :  SM  : 
MG  ;  A  is  the  perihelion. 

To  find  the  aphelion,  produce  KS  to  B,  so  as  to  make  SB  : 
LK  : :  SM  :  MG  ;  B  is  the  aphelion. 

Bisect  AB  in  C ;  then  SO  divided  by  AC  is  the  eccentricity 
of  the  orbit. 

The  longitude  of  the  perihelion  is  known  from  the  angle 
MSA,  already  obtained ;  for  the  longitude  of  SM  is  given  at 
the  outset. 

333.  Masses  of  bodies  compared  ~by  the  orbits  described, 
about  them. — The  mass  of  a  body,  whether  the  sun  or  a  planet, 
can  be  compared  with  that  of  another,  by  means  of  the  distance 
and  period  of  a  planet  or  satellite  revolving  about  each.  It 
has  been  proved  (Chap.  VIII.)  that  gravity  varies  directly  as 
the  mass,  and  inversely  as  the  square  of  the  distance ;  that  is, 

G  oo  Y~ .     It  was  shown,  also  (Chap.  VI.),  that  the  centripetal 


172  MASSES   OF  BODIES   COMPAEED. 

force  or  gravity  varies  directly  as  the  distance,  and  inversely 
as  the  square  of  the  periodic  time  ;  that  is,  G  oo  -p-.  There- 

fore, ^  oo  -^r  ;  /.  M  GO  p^-  ;    or,   the  mass   of  the   central 

body  varies  directly  as  the  cube  of  the  distance,  and  inversely 
as  the  square  of  the  periodic  time  of  the  body  revolving  about 
it.  Thus,  to  compare  the  mass  of  the  sun,  about  which  the 
earth  revolves,  and  the  mass  of  the  earth,  about  which  the 
moon  revolves,  we  have 


650)3     (95,134,000)' 

32?    :     P5-.256T  '•  :  l  : 
Therefore,  the  mass  of  the  sun  is  about  354,000  times  as  great 
as  that  of  the  earth. 

334.  Examples.  — 

1.  Were  the  earth's  mass  equal  to  the  sun's,  in  what  time 
would  the  moon,  at  its  present  distance,  revolve  about  it  ? 
Letting  x  stand  for  the  time  required,  we  have  1  :  354,000  :  : 

I3         I3 
•^^  :  -=-.     Ans.  Ih.  6m.  7s. 


(27.32)2 

2.  How  much  must  the  mass  of  the  earth  be  increased, 
in  order  that  the  moon  may  revolve  about  it  in  the  same 
time  as  it  now  does,  when  removed  to  three  times  its  present 
distance  \     Ans.  It  must  be  27  times  as  great. 

3.  The   distance  of  Jupiter  from    the  sun   is   496,000,000 
miles,  and  its  periodic  time  is   4332.585  days.     The  fourth 
satellite  is   1,200,000  miles   from  the  primary,  and  revolves 
in  16d.  16h.  31m.     Compare  the  mass  of  the  sun  with  that  of 
Jupiter.     Ans.  1048  :  1. 

4.  The  moon  revolves  in   27.32   days,  at  the   distance  of 
238,650  miles  from  the  earth  ;   Jupiter's  second  satellite  re- 
volves in  3.552  days,  at  the  distance  of  442,900  miles.     What 
are  the  relative  masses  of  the  earth  and  Jupiter  ? 

Ans.  1  :  378. 

335.  Masses  of  planets  which   have    no  satellites. — The 
method  described  in  the  preceding  article  can  be  applied  to 


PERTURBATIONS   OF   THE   PLANETS.  173 

the  sun,  and  all  those  planets  which  are  attended  by  satellites. 
But  Mercury,  Yenus,  and  Mars,  which  have  no  satellites,  must 
be  compared  in  some  other  way.  Each  of  these  planets,  by  its 
attraction,  sensibly  disturbs  the  motion  of  the  planet  nearest  to 
it ;  and  the  degree  of  this  disturbance,  the  distance  being 
known,  is  a  measure  of  its  quantity  of  matter.  Thus,  the 
masses  of  Venus  and  Mars  can  each  be  estimated,  by  observing 
the  force  which  they  exert  on  the  earth  when  passing  near  it. 
The  mass  of  Mercury  has  been  determined  by  its  disturbing 
power  exerted  on  Encke's  comet,  as  well  as  on  the  planet 
Venus. 

336.  Densities  of  the  planets. — The  masses  of  bodies  vary 
as   the   products  of  their  volumes  and  densities.     Therefore 
their  densities  vary  as  the  masses  divided  by  the  volumes. 
The  densities,  as  given  in  Table  IV,  may  be  obtained  in  this 
way,  and  reduced  to  a  scale,  in  which  the  earth's  density  is 
called  1.     Or  they  may  be  reduced  to  a  scale,  in  which  the 
density  of  water  is  1 ;  in  the  last  form,  the  numbers  are  called 
specific  gravities.     These  also  are  given  in  Table  IV. 

337.  Perturbations  of  the  planets. — The  solar  system,  as 
we  have  seen,  consists  of  many  bodies ;  and  each  one  of  them 
attracts  every  other  one,  and  attracts  it  more,  according  as  it 
is  nearer  and  more  massive.     Hence,  no  planet  can  continue 
to  pursue  the  same  elliptic  orbit  about  the  sun,  as  if  the  sun 
and  planet  were  the  only  bodies.     Nor  can  any  satellite  de- 
scribe its  orbit  undisturbed  about  the  primary.     The  number 
and  variety  of  these  disturbing  forces  exerted  within  the  system 
are  very  great.  But  many  of  them  are  so  minute  as  to  be  in- 
sensible.    As  was  shown  in  Chapter  X.  respecting  the  moon, 
so  in  regard  to  every  planet  and  satellite,  the  disturbing  influ- 
ences are  of  various  kinds,  some  tending  to  alter  the  plane  of 
the  orbit,  others  to  change  its  form,  etc.     These  perturbations, 
like  those  of  the  moon,  are  classed  into  periodical  and  secular. 

338.  Retrogradation  of  nodes. — If  the  orbit  of  a  planet  is 
oblique  to  that  of  another,  one  component  of  the  disturbing 
force  tends  to  move  the  nodes  of  the  two  orbits  backward,  as 


174         PERTURBATIONS  OF  THE  PLANETS. 

shown  in  Art.  192.  And  every  satellite,  whose  orbit  is  inclined 
to  that  of  its  primary,  is  acted  on  by  the  sun,  in  the  same  man- 
ner as  the  moon  is :  its  nodes  retrograde  on  the  orbit  of  the 
primary.  For  the  planets,  this  retrograde  motion  is  excessively 
slow,  generally  amounting  to  only  a  few  minutes  in  a  century. 

339.  Change  of  inclination. — Another  disturbance  of  the 
orbit  of  a  planet  has  respect  to  its  inclination  to  the  orbit  of 
another.     There  are  small  periodical  oscillations  in  the  inclina- 
tion, at  every  revolution,  which  nearly  compensate  each  other, 
like  those  of  the  moon's  orbit  (Art.  193).     But  the  compensa- 
tion not  being  exact,  there  is  a  minute  change,  which  remains 
unbalanced,   and  accumulates  for  many  centuries,   when    the 
change  is  reversed  and  accumulates  in  the  opposite  direction. 
These   secular    oscillations,   however,   are    all   within    narrow 
limits.     Thus,  the  ecliptic,  though  generally  spoken  of  as  a 
fixed  plane,  is  not  truly  so,  but  is  subject  to  a  minute  change 
of  a  few  seconds  in  a  century.     It  is  proved  that  the  whole 
variation  can  never  amount  to  3°,  and  that  within  that  range 
it  will  occupy  many  thousands  of  years  in  making  a  single  sec- 
ular oscillation. 

340.  Advance  of  apsides. — All  planets  within  the  orbit  of 
a  given  planet,  conspire,  on  the  whole,  to  increase  its  gravity 
toward  the.  sun  ;  while  the  general  effect  of  those  outside  of 
the  same  orbit  is  to  diminish  it.     It  was  shown  (Art.  183)  that 
the  sun,  being  outside  of  the  moon's  orbit  about  the  earth, 
sometimes  increases  and  sometimes  diminishes  the  moon's  ten- 
dency to  the  earth,  but  on  the  whole  diminishes  it.     The  same 
thing  is  true  of  every  planet  outside  of  the  orbit  of  another. 
One  consequence  of  diminished  attraction  is,  to  cause  the  line 
of  apsides  to  advance  and  to  retrograde  alternately ;   but  the 
resultant   of  the   whole   action   is   an   advance.     The   earth's 
apsides  advance  11  £"  in  a  year  (Art.  147).     So  the  line  of 
apsides  of  most  of  the  planetary  orbits  has  a  slow  motion  from 
west  to  ea?t. 

341.  Change  (f  eccentricity. — A  planet  tends  to  increase 
the   eccentricity  of  an   orbit  within  its  own,  when  the  two 


PERTUKBATIONS    OF    THE    PLANETS.  175 

planets  are  in  its  line  of  apsides  at  conjunction  and  opposition, 
and  to  diminish  it  when  the  line  from  the  sun  to  the  two  plan- 
ets makes  a  right  angle  with  the  line  of  apsides;  analogous  to 
the  action  of  the  sun  on  the  moon's  orbit  (Art.  1ST).  These 
disturbances  are  very  minute,  but  they  will  not  balance  each 
other  during  a  sy nodical  revolution  ;  and  therefore  there  is  a 
small  secular  change  in  the  eccentricity  of  the  orbits.  For  ex- 
ample, the  eccentricity  of  the  earth's  orbit  has  been  diminish- 
ing, and  for  many  thousands  of  years  to  come  it  will  continue  to 
diminish,  at  the  rate  of  O.OOOOi  of  its  value  per  century.  The 
orbit,  however,  will  never  reach  the  exact  form  of  a  circle,  but 
after  arriving  to  a  minimum  of  eccentricity,  it  will  begin  to 
return  to  a  more  eccentric  form,  and  thus  will  oscillate  about  a 
mean  value  perpetually.  And  the  range  of  its  eccentricity  is 
so  limited,  that  the  ellipse,  if  correctly  represented,  can  never 
differ  visibly  from  a  circle. 

It  is  this  slow  change  in  the  earth's  orbit  which  causes  the 
secular  inequality  of  the  moon's  motion  (Art.  195). 

342.  Change  in  the  length  of  the  major  axis. — There  are 
also  minute  periodic  changes  in  the  length  of  the  major  axis  of 
an  orbit ;  that  is,  in  the  mean  distance  of  a  planet  from  the 
sun.     But  both  calculation  and  observation  establish  the  fact, 
that  there  is  no   secular  inequality,    because   the   periodical 
changes  exactly  compensate  each  other.     And,  if  the  mean  dis- 
tance of  each  planet  from  the  sun  has  permanently  the  same 
value,  then,  according  to  Kepler's  third  law,  the  jjeriodic  time 
is  also  constant. 

343.  Long  periods. — There  are  in  the  solar  system  several 
cases  of  inequality,  accumulating  for  centuries,  which  never- 
theless have  the  character  of  periodical  rather  than  secular 
inequalities,  and  depend  on  the  fact  that  the  periodic  times  of 
two  planets  almost  exactly  measure  a  certain  length  of  time. 

For  example,  the  earth  makes  8  revolutions  in  very  nearly 
the  same  time  in  which  Venus  makes  13.  Hence,  every  fifth 
conjunction  occurs  within  1|°  of  the  same  points  of  their  re- 
spective orbits.  At  the  end  of  this  period  of  8  years,  there  is 
a  minute  perturbation,  which  remains  uncompensated,  and 


176         PERTURBATIONS  OF  THE  PLANETS. 

which  is  about  doubled  at  the  end  of  16  years,  anrl  tripled  at 
the  end  of  21  years,  and  so  on.  This  disturbance  is  very  small, 
never  amounting  to  more  than  a  few  seconds ;  but  it  requires  a 
period  of  240  years  in  order  to  pass  through  all  its  values. 

The  long  inequality  of  Jupiter  and  Saturn  is  a  more  remark- 
able case.  Jupiter  makes  5  revolutions,  and  Saturn  2,  in 
nearly  the  same  time.  An  unbalanced  disturbance,  which  ap- 
pears at  the  end  of  this  time,  goes  on  accumulating.  During 
the  17th  century,  Saturn  was  constantly  retarded,  and  Jupiter 
accelerated.  But  in  the  18th,  this  was  reversed,  and  Saturn  is 
now  accelerated,  and  Jupiter  retarded.  This  will  continue  still 
longer ;  and  the  whole  period  required  for  this  inequality  is 
more  than  900  years.  The  deviation,  at  its  maximum,  is  49' 
for  Saturn  and  21 '  for  Jupiter. 

344.  Degree  of  dicing e  in  the  several  elements. — Of  the  sev- 
eral elements  named  at  the  beginning  of  this  chapter,  we  see, 
from  what  precedes,  that  the  following  classification  may  be 
made. 

1.  The    1st   and   2d  have   no   secular  inequality  whatever. 
Their  value  remains  constant  from  age  to  age.     The  perma- 
nency of  these  two  elements  secures  a  constant  length  of  the 
year,  and  a  constant  amount  of  heat  from  the  sun  on  each 
planet. 

2.  The  3d  and  6th  elements  have  small  periodical  oscilla- 
tions, but  their  secular  change  is  in  one  direction ;  the  nodes 
perpetually  retrograde,  the  apsides  perpetually  advance.     But 
the  continual  change  in  the  same  direction  in  these  two  ele- 
ments has  no  tendency  to  derange  the  condition  of  things  on  a 
planet.     As  to  the  well-being  of  the  occupants  of  a  planet,  it  is 
of  no  consequence  how  the  major  axis  of  its  orbit  is  situated, 
if  only  the  form  of  the  ellipse  is  preserved.     It  is  also  im- 
material in  what  direction  the  line  of  its  nodes  may  happen 
to  lie. 

3.  The  4th  and  5th  elements  have  both  periodical  and  secu- 
lar inequalities;    but  they  range  within   very  narrow  limits. 
The  smallness  of  these  changes  insures  all  the  planets  against 
any  considerable  change  from  year  to  year,  in  respect  to  the 
extremes  of  heat  and  cold,  and  in  respect  to  the  seasons. 


STABILITY   OF   THE   SYSTEM.  177 

345.  Stability  of  the  system. — Several  of  the  secular  in- 
equalities, before  their  true  character  had  been  demonstrated, 
excited  great  interest  among  astronomers,  because  they  seemed 
to  indicate  tendencies  toward  the  ultimate  destruction  of  the 
solar  system.     If  the  eccentricity  of  the  earth's  orbit  should 
continue  to  change  in  the  same  direction  perpetually,  the  earth 
would  at  length,  though  perhaps  not  in  millions  of  years,  be- 
come unfit  to  be  the  habitation  of  man,  because  of  the  terrible 
extremes  of  heat  and  cold  at  perihelion  and  aphelion.     So,  if 
the  inclination  of  equator  and   ecliptic   should   continue  its 
change  perpetually  in  the  same  direction  as  at  present,  the  sea- 
sons would  by  and  by  disappear,  and  afterward  run  to  an  ex- 
treme which  would  produce  desolation  over  the  whole  surface 
of  the  earth.     And  if  the  secular  inequality  of  the  moon's 
period  were  always  to  go  on  as  it  has  done  for  centuries  past, 
the  moon  would  at  length  be  precipitated  on  the  earth. 

But  La  Grange,  La  Place,  and  others  have  demonstrated 
that  all  the  perturbations  have  their  limits,  and,  after  in- 
creasing with  extreme  slowness  for  many  ages,  must  again  de- 
crease in  like  manner ;  and,  furthermore,  that  the  entire  series 
of  changes  lies  within  so  narrow  bounds,  that  no  disastrous 
consequences  can  ensue. 

346.  Manner  in  which  the  stability  is  secured. — The  sta- 
bility of  the   system  is  secured  by  the  fulfilment  of  certain 
essential  conditions  in  the  arrangement  of  its  parts. 

1.  The  great  mass  of  matter  constituting  the  solar  system  is 
in  the  central  body,  the  sun  being  700  times  as  great  as  all  the 
other  bodies  united.     Hence,  all  movements  are  principally 
controlled  by  the  sun. 

2.  The  planets,  and  especially  the  larger  ones,  are  at  great 
distances  from  each  other ;  and  thus  the  sun's  influence  over 
each  is  but  little  modified  by  their  mutual  attractions. 

3.  The  orbits,  especially  of  the  largest  planets,  have  but 
slight  eccentricity,  and,  therefore,  always  maintain  their  great 
distances  from  each  other. 

4.  The  mutual  inclinations  of  the  orbits  are  small.     Hence, 
there  are  no  large  forces  operating  to  change  the  position  of 
orbits,  and  thus  disturb  the  seasons. 

12 


178  RELATIONS   OF   ELEMENTS. 

347.  Relations  of  the  planetary  motions*  —  The  planets  are 
so  adjusted  to  each  other,  in  respect  to  their  velocities,  dis- 
tances from  the  sun,  and  gravity  toward  the  sun,  that  if  any 
one  of  these  relations  between  two  planets  is  known,  all  the 
others  become  known  also. 

Let  T  —  mean  distance  ;   t  =  periodic  time  ;   v  =  velocity  ; 

g  =  gravity.     Also,  let  s  (slowness)  =  —  ,  the  reciprocal  of  ve- 
locity ;  and  I  (lightness)  =  —  ,  the  reciprocal  of  gravity. 

>  7  (Art.  92); 

T1 

o°  -3--     But;  by  Kepler's  third  law, 
t 

r*  1 

f  oo  V  :  .*. 


As  s  =  —  ,  .*.  v  =  —  «,  and  -0   =  -=-  ;  /.  —  oo  —  ,  and  6-  GO  r. 
fr  «  -  s2  '       «"        r  ' 

r  rs  f 

Again,  since  v  oo  —  ,  v3  GO  -5-.     But,  r3  oo   f  ;  /.  Vs  QO  —  , 

V  V  V 

or  Vs  oo  —  ;  /.  ~Y  oo  —  ,  and  s3  oo  t. 
t  '       ,93          ^  ' 

By  the  law  of  gravity,  g  oo  —  ;  .*.  —  GO   —  ;  and  I  QO  r*. 

But  sa  oo  /•;  /.  ^4  QO  r\  and  6-4  oo  I. 

Bringing  together  these  results,  we  find  four  variations, 
s  oo  s1  ;  r  oo  s2  ;  £  oo  s3  ;  I  GO  s4. 

Hence,  we  have  the  reciprocal  of  velocity,  s  j  the  distance, 
r  j  the  periodic  time,  t  ;  and  the  reciprocal  of  gravity,  I  /  re- 
spectively denoted  in  their  ratios  by  the  geometrical  series,  s1, 
s2,  5s,  s4,  in  which  the  first  term  and  the  ratio  are  equal. 

348.  Mode  of  using  these  variations  for  calculation.  —  If 
the  velocities  of  two  planets  are  given,  we  first  take  their  re- 
ciprocals, and  thus  have  the  ratio  of  s  for  the  two.  The  terms 
of  this  ratio  are  then  raised  to  the  second,  third,  or  fourth 
power,  according  as  we  wish  to  compare  r,  or  t,  or  I. 


RELATIONS   OF    ELEMENTS.  179 

But  if  the  ratio  of  distances,  or  times,  or  gravities  is  given, 
the  corresponding  root  is  first  extracted,  in  order  to  find  the 
ratio  in  respect  to  *,  and  then  we  proceed  as  before. 

349.  Examples  — 

1 .  The  planetoid  Pallas  has  a  period  of  4f  years ;  how  much 
further  is  it  from  the  sun  than  the  earth  is  1     How  much  less  is 
it  attracted  ?     How  much  slower  does  it  move  ? 

Let  #,  s,  r,  I  be  used  for  the  earth,  and  T,  S,  K,  L  for  Pallas. 

Then,      .  t :  T  ::  1  :  4.667; 

.-.  I3'  :  (4.667)*  ::  *  :  S; 
/.  s  :  S  : :  1  :  1.67  ;  that  is,  the  earth's  ve- 
locity is  1.67  times  as  great  as  that  of  Pallas. 

Again,  r  :  E  : :  I2  :  (1.67)2  : :  1  :  2.7926  ;  or,  Pallas  is  2.7920 
times  as  far  from  the  sun  as  the  earth  is. 

Again,  / :  L  : :  I4 :  (1.67)4  : :  1  :  7.7985  ;  therefore,  the  earth  is 
attracted  by  the  sun  about  7.8  times  as  much  as  Pallas  is. 

2.  What  would  be  the  period  of  a  satellite  revolving  about 
the  earth  close  to  its  surface  ? 

The  distance  of  this  satellite  to  that  of  the  moon  is  as  1  :  60 ; 

.-.  s  :  S  : :  1  :  (60;* ;  .-.  t :  T  : :  1  :  (60)*  : :  1  :  464.66. 

But  the  moon's  period  is  27.32  days,  or  655.68  hours. 
Hence,  the  period  of  the  satellite  is  1.411  h.,  or  Ih.  24m.  30s. 
nearly. 

3.  How  much  faster  must  the  earth  rotate  on  its  axis,  in 
order  that  bodies  on  the  equator  may  lose  all  their  weight  ? 

This  is  just  the  condition  of  the  body  in  example  2d,  whose 
period  is  1.41  Ih.  But  the  earth's  time  of  rotation  is  2±h., 
which  is  17  times  1.411h.  Therefore,  if  the  earth  were  to 
rotate  17  times  more  rapidly  than  at  present,  all  bodies  on 
the  equator  would  just  lose  their  weight,  and  revolve  inde- 
pendently. 

4.  What  would  be  the  periodic  time  of  a  body  revolving 
about    the   earth,    at   the   distance   of  5,000   miles   from   the 
center?  Ans.  Ih.  59m. 


180  COMETS. 

5.  What  must  be  the  moon's  distance  from  the  earth,  in 
order  to  revolve  about  it  once  in  a  year  ? 

Ans.  1,344,000  miles. 

6.  Suppose  a  planet  to  be  discovered,  whose  daily  velocity  is 
5  times  as  great  as  that  of  Mercury,  what  is  its  distance  from 
the  sun's  center  ?     Ans.  1,480,000. 


CHAPTER  XYIII. 

COMETS. — SHOOTING  STARS. 

350.  A  comet  defined. — A  comet  is  a  body  which  consists 
of  nebulous  matter,  and  revolves  about  the  sun  in  a  very  eccen- 
tric orbit.     Most  comets  present  a  roundish  ill -defined  appear- 
ance, often  having  a  bright  central  part  called  the  nucleus. 
The  fainter  part,  surrounding  the  nucleus,  is  called  the  coma 
(hair) ;    and  the   tail,  which   distinguishes  many   comets,  is 
merely  the  extension   of  the  coma.     It  is  the  streaming  ap- 
pearance of  the  tail,  resembling  hair,  which  gave  the  name 
"  comet"  to  this  class  of  bodies.     The  nucleus  has  been  some- 
times supposed  to  be  solid  ;  but  it  probably  consists  always  of 
nebulous   matter  in  a  more  condensed  state  than  the  other 
parts.      The   nucleus   and  coma   are  called  the   head  of  the 
comet. 

351.  Number  of  comets. — Many  hundreds  of  comets  have 
been  recorded,  most  of  them,  of  course,  visible  to  the  naked 
eye.     But  lately  it  is  observed  that  most  comets  are  telescopic 
objects.     And  many  which  would  otherwise  be  seen,  escape 
observation  by  being  above  the  horizon  only  in  the  daytime. 
The  whole  number,  therefore,  belonging  to  the  solar  system 
is  undoubtedly  to  be  reckoned  by  thousands,  or  tens  of  thou- 
sands. 

352.  Eccentricity  of  orbit. — All  known  cometary  orbits  are 
more  eccentric  than  any  planetary  orbit ;  and  most  of  them  are 


ECCENTRIC   DEBITS.  181 

exceedingly  so,  their  perihelion  being  as  near  the  sun  as  Mer- 
cury and  Yen  us,  or  nearer,  and  their  aphelion  as  far  off  as  the 
most  distant  planets,  or  even  beyond.  And  some  appear  to  be 
ellipses  of  infinite  length — that  is,  parabolas ;  while  others  ex- 
hibit the  form  of  hyperbolas.  In  orbits  of  these  last  forms, 
comets  can,  of  course,  pass  the  perihelion  but  once. 

353.  Conseq uences  of  great  eccentricity. — 

1.  One  effect  of  this  great  eccentricity  is,  that  a  comet  is  too 
far  from  the  earth  to  be  seen,  except  during  a  small  part  of  its 
revolution,  while  it  is  near  the  center  of  the  system. 

2.  Another  effect  is,  that  great  changes  take  place  in  the 
condition  of  the  nebulous  matter  of  which  the  comet  is  com- 
posed.    As  a  comet  approaches  the  sun,  both  the  nucleus  and 
coma  grow  less  in  diameter,  and  enlarge  again  as  it  departs. 
But  the  tail,  if  there  is  one,  is  rapidly  lengthened  as  the  comet 
approaches,  and  is  diminished  in  length  when  it  withdraws. 
Sometimes,  a  comet,  whose  appearance  is  spherical  when  first 
seen,  begins  suddenly  to  exhibit  the  formation  of  a  tail  as  it 
comes  nearer,  which  at  length  stretches  over  a  large  arc  of  the 
sky ;  and  after  the  perihelion  passage,  as  it  departs  from  the 
sun,  the  tail  wholly  disappears  before  the  comet  becomes  in- 
visible. 

It  might  be  supposed  that  this  diminution  of  the  coma  re- 
sults from  the  loss  of  material  which  is  taken  away  to  form  the 
tail,  w^hile  the  comet  is  approaching  the  sun  ;  and  that  the  sub- 
sequent enlargement  is  due  to  the  return  of  the  same  material, 
as  the  tail  is  contracted.  But  this  will  not  fully  explain  the 
observed  changes;  for  the  contraction  and  subsequent  expan- 
sion occur  when  no  tail  is  formed.  Hence,  it  is  supposed  that 
the  heat  of  the  sun  reduces  the  dimensions  of  the  nucleus  by 
expanding  a  portion  of  it  into  the  coma,  and  also  changes  the 
nebulous  matter  of  the  coma  into  a  pure,  transparent  gas,  which 
is  afterward  condensed  into  a  visible  form  again,  as  the  comet 
withdraws  .from  the  sun. 

354.  Form  and  direction  of  tails  of  comets. — The  forms  of 
tails  belonging  to  different  comets  are  exceedingly  varied.     In 
general,  however,  the  sides  diverge  from  the  head,  so  that  the 


182 


TAILS   OF   COMETS. 


most  distant  and  faintest  part  is  broadest,  as  in  the  comets  of 
1680  and  1811  (Figs.  82,  83).  In  some  cases,  the  divergency  is 
very  slight,  as  in  the  comet  of  1843  (Plate  I.,  at  the  end). 


Fig.  82. 


Fig.  83. 


COMET  OF  1811. 


COMET  OP  1680. 


ISTot  infrequently,  the  principal  light  of  the  tail  appears  to 
proceed  from  its  edges,  presenting  somewhat  the  aspect  of  two 
tails  diverging  from  the  sides  of  the  coma.  In  such  cases,  the 
coma  and  tail  seem  to  have  the  form  of  a  hollow  paraboloid, 
so  that  we  look  through  a  much  greater  extent  of  illuminated 
matter  on  the  sides  than  in  the  central  parts.  In  the  comet  of 
1858,  the  nucleus  was  at  one  time  surrounded  by  a  series  of 
parabolipnl  envelopes,  which  increased  in  number  as  the  comet 
approached  the  sun.  (PL  II.,  Fig.  1). 

In  a  few  instances,  the  tail  has  been  known  to  consist  of  sev- 
eral luminous  rays,  diverging  from  each  other,  as  the  comet  of 
1744,  in  which  tluere  were  six,  the  extreme  ones  making  with 
each  other  an  angle  of  about  45°. 


DIMENSIONS  OF  'COMETS.  183 

The  general  direction  of  the  tail  is  from  the  sun'  so  that,  as 
a  comet  approaches  the  sun,  the  tail  follows  it ;  but  as  it  re- 
cedes, the  tail  is  directed  forward.  The  axis  of  the  tail  is  not, 
however,  a  straight  line,  but  more  or  less  curved  backward,  so 
that  the  convex  side  of  the  curve  is  foremost  in  the  motion. 

355.  Cause  of  the  direction  and  curvature  of  the  tail. — 
Modern  telescopic  observations  on  some  of  the  most  conspicu- 
ous comets,  show  that  the  material  of  which  the  tail  is  formed 
is  first  projected  toward  the  sun,  rather  than  from  it ;  and  that 
some  force  emanating  from  the  sun  then  drives  it,  with  great 
velocity,  in  the  opposite  direction,  causing  it  to  sweep  past  the 
nucleus  on  both  sides,  and  stretch  millions  of  miles  into  space. 
The  rate  at  which  it  is  thus  driven  from  the  sun  is  sometimes 
enormous.     In  the  case  of  Halley's  comet,  in  1835,  the  nebu- 
lous matter  had  a  velocity  of  2,000,000  miles  per  day..    In  Do- 
nati's  (1S58),  it  reached  the  rate  of  8,000,000  miles  per  day. 
What  force  it  is  which  the  sun  thus  exerts  in  a  direction  oppo- 
site to  its  gravity,  it  is  vain  to  conjecture.   It  must  be  supposed 
that  at  least  a  part  of  the  material  driven  so  violently  from  the 
comets  is  dissipated  and  lost ;  and  there  is  indication  of  this  in 
the  diminished  size  and  brilliancy  of  those  wiiose  returns  have 
been  noticed.     Perhaps  the  numerous  comets  which;  have  no 
tails  have  been  divested  of  them  by  this  process. 

The  bending  of  the  tail  backward  is  a  necessary  consequence 
of  the  longer  arc  which  the  extreme  part  of  the  tfil  must  de- 
scribe. The  material  of  the  tail  has  the  same  velocity  in  the 
orbit  as  the  head,  when  it  is  driven  from  it.  This  velocity  it 
retains ;  but,  having  to  describe  a  curve  about  the  sun:  several 
millions  of  miles  outside  of  the  other,  it  must,  of  course^  fall 
behind  it. 

356.  Dimensions  of  comets. — The  dimensions  of  comets-  are 
various,  and,  on  account  of  their   nebulous   character,  they 
never  admit  of  accurate  measurement.     The  nucleus  of  a.. large 
comet  is  sometimes  5,000  miles,  and  the  coma  200,000,  miles 
in  diameter,  while  the  tail  has,  in  one  case,  attained  the  extra- 
ordinary length  of  200,000,000  miles. 

The  apparent  length  of  a  comet's  tail  is  often  sufficient  to 


184  DIRECTIONS    OF    COMETARY    MOTIONS. 

span  an  arc  of  20°  or  30°  on  the  sky,  and  sometimes  much 
more  than  this.  The  comet  of  1680  extended  97°,  and  that  of 
1861,  106°.  The  fainter  part,  in  all  cases,  is  seen  only  by  in- 
direct vision. 

It  is  obvious  that  the  real  length  can  not  be  inferred  from  the 
apparent,  until  the  distance  from  us,  and  the  obliquity  to 
our  line  of  vision,  are  obtained. 

357.  Light  of  the  comets. — These  bodies,  like  the  planets 
and  satellites,  shine  by  solar  light  which  they  reflect  to  us. 
But,  unlike  all  planetary  bodies,  they  are  in  a  condition  so 
attenuated,  that  the  sun's  rays  penetrate  every  part  of  them 
without  obstruction.    The  brightness  of  a  star  is  not  diminished 
in  the  least  when  seen  through  the  tail  or  coma  of  a  comet.    In 
a  few  instances,  a  star  has  been  seen  through  the  nucleus,  and 
even  then  was  not  essentially  dimmed. 

A  satisfactory  proof  that  the  comets  are  seen  by  the  sun's 
light  which  they  reflect,  is,  that  their  brightness  diminishes  as 
they  recede  from  the  sun ;  so  that  they  are  at  length  lost  to 
view,  not  by  being  too  small  to  fill  an  appreciable  angle,  but 
too  faint  to  be  visible.  This  would  not  be  true  of  a  self-lumi- 
nous body :  its  brightness  would  remain  the  same  at  all  dis- 
tances from  us  ;  that  is,  its  light  would  diminish  no  faster  than 
its  apparent  area. 

358.  Quantity  of  matter  in  comets. — Though  some  of  the 
largest  comets  surpass  all  other  bodies  in  the  solar  system  in 
magnitude,  yet  in  respect  to  their  mass  they  are  too  small  to 
have  produced  as  yet  the  slightest  perceptible  effect.     They 
sometimes  come  very  near  planets  and  their  satellites,  but  are 
never  known  to  exert  the  least  influence  on  them.     They  do, 
of  course,  attract  the  planets,  because  they  are  attracted  by 
them,  and  .suffer  great  disturbances  from  them.      But  until 
they  themselves  produce  some  effect  which  is  appreciable,  their 
mass  must  be  regarded  as  infinitely  small. 

359.  Directions  of  cometary  motions. — The  cometary  orbits 
are  unlike  the  planetary,  not  only  in  the  degree  of  their  eccen- 
tricity, but  in  the  varied  positions  of  their  planes.     Instead  of 


ORBITS    OF    COMETS.  185 

being  limited  to  a  narrow  zone  like  the  zodiac,  they  make 
every  variety  of  angle  with  the  ecliptic,  so  that  a  comet  is  as 
likely  to  pass  round  the  sun  from  north  to  south  as  from  west 
to  east.  And  whether  the  orbit  is  much  or  little  inclined,  the 
comet's  motion  in  it  is  as  often  retrograde  as  direct. 

360.  Means  of  determining  a  comet's  orbit. — Since  a  comet 
can  be  seen  only  during  the  time  of  its  describing  a  short  arc 
near  the  perihelion,  the  astronomer  has  not  the  same  oppor- 
tunity for  fixing  its  orbit  as  he  has  in  the  case  of  a  planet, 
which  can  be  observed  in  all  parts  of  its  course. 

It  is  true  in  theory,  that  by  any  three  observations  on  the 
position  of  a  body  revolving  about  the  sun,  its  whole  orbit  can 
be  determined.  But  if  it  is  very  eccentric,  and  the  observa- 
tions are  confined  to  a  small  portion  at  one  extremity,  the 
slightest  error  may  greatly  change  the  distance  of  the  aphelion, 
and  consequently  the  length  of  the  axis  and  the  periodic  time. 

It  is  usual,  therefore,  to  assume  the  path  to  be  a  parabola — 
that  is,  an  ellipse  of  infinite  length,  whose  eccentricity  is  1. 
There  are  then  \m£fowr  of  the  seven  elements  (Art.  325)  to  be 
determined — namely,  the  3d,  4th,  6th,  and  7th.  But  instead  of 
the  2d  element  in  Art.  325,  there  may  be  substituted  the  peri- 
helion distance,  making  in  all  five  elements,  as  follows  : 

2.  The  perihelion  distance. 

3.  The  longitude  of  the  ascending  node. 

4.  The  inclination  of  its  orbit  to  the  ecliptic. 

6.  The  longitude  of  the  perihelion. 

7.  The  place  of  the  body  at  a  given  time. 

These  five  elements  may  usually  be  determined  without 
much  difficulty. 

361.  Process    of  finding   the  five    elements. — The  right 
ascension  and  declination  are  observed  each  night  with  all 
possible  care,  and  the  exact  time  of  every  observation  is  re- 
corded.    Three  of  these  dates  are  selected,  several  days  apart ; 
and  from  the  right  ascension  and  declination  at  each  date  are 
deduced,  first  the  geocentric,  and  thence  the  heliocentric  longi- 
tude and  latitude  of  the  comet.     The  heliocentric  places  of  the 
earth  at  the  same  times  are  known,  since  each  is  180°  from  the 


186  COMETAKY    ELEMENTS. 

sun's  apparent  place  at  the  same  time.  If  we  imagine  three 
straight  lines  to  be  drawn  from  the  known  places  of  the  earth 
through  the  corresponding  positions  of  the  comet,  its  distance 
from  us  in  each  line  must  be  determined  by  the  following  con- 
ditions, in  accordance  with  Kepler's  laws : 

1.  A  plane  passing  through  the  three  positions  must  also 
pass  through  the  sun. 

2.  The  three  places  must  be  in  a  parabola,  whose  focus  is  at 
the  sun. 

3.  The  areas  included  between  the  radii  vectores  drawn  to 
the  sun  must  be  proportional  to  the  times. 

Points  are  successively  assumed  in  the  given  lines,  until  at 
length  those  are  found  which  w^ill  fulfill  the  above  conditions. 
By  this  tentative  process  the  orbit  is  approximately  deter- 
mined. 

Three  other  dates  may  then  be  tried  in  the  same  manner ; 
and  if  the  results  nearly  agree,  the  mean  may  be  considered 
more  accurate  than  either. 

This  method  is  just  as  applicable  for  determining,  approx- 
imately, the  orbit  of  a  newly  discovered  planet  or  planetoid. 
But  in  these  cases,  the  observations  can  usually  be  followed 
up  in  various  parts  of  the  orbit,  and  thus  previous  errors  cor- 
rected. 

362.  Determination  of  the  remaining  elements. — If  a  comet 
remains  in  sight  for  several  months,  it  is  quite  probable  that 
long-continued  and  careful  observations  will  show  that  the  orbit 
is  not  truly  a  parabola,  but  an  ellipse.  In  such  a  case,  the  1st 
and  2d  elements  (Art.  325)  may  be  computed,  and  the  5th 
corrected.  Though  there  may  be  good  evidence  that  the  orbit 
is  an  ellipse,  yet  great  uncertainty  must  attach  to  any  determi- 
nation of  periodic  time,  mean  distance,  and  eccentricity,  until 
they  are  settled  conclusively  by  a  return  of  the  comet  to  its 
perihelion. 

A  comet,  on  its  return  to  its  perihelion,  is  not  to  be  identified 
so  much  by  its  physical  aspect,  as  by  the  agreement  of  its  ele- 
ments. If  the  place  of  the  node,  the  inclination  to  the  ecliptic, 
the  place  of  the  perihelion,  and  the  perihelion  distance  agree 
very  nearly  with  those  of  some  comet,  which  has  been  previ- 


187 

ously  seen,  it  is  fairly  presumed  to  be  the  same,  even  though  its 
appearance  may  essentially  differ. 

When  a  comet  is  thus  identified,  its  periodic  time  is,  of 
course,  known ;  and  from  this,  its  mean  distance,  and  the  ec- 
centricity of  its  orbit  are  readily  obtained. 

363.  Comets  whose  elements  have  been  computed,  ~but  not 
verified. — The  orbits  of  more  than  200  comets  have  been  com- 
puted. But  a  great  majority  of  these  appeared  to  be  parabolas, 
and  no  prediction  of  their  return  could  be  made.  In  40  or  50 
cases,  the  movement  of  the  comet  seemed  to  afford  evidence 
that  the  orbit  was  an  ellipse,  and  in  7  others,  a  hyperbola. 
For  the  elliptic  orbits,  the  returns  \vere,  of  course,  predicted. 
But  most  of  the  computed  periods  are  long,  generally  hundreds, 
and,  in  several  instances,  thousands  of  years  ;  so  that  as  yet 
very  few  of  them  have  been  verified.  The  period  of  a  comet 
seen  in  1849  was  calculated  to  be  2,115  years.  If  the  compu- 
tation be  supposed  correct,  the  distance  of  this  comet  from  the 
sun  at  aphelion  is  about  11  times  the  distance  of  Neptune,  and 
its  next  return  will  occur  in  the  year  3964.  But  the  results  of 
calculation,  in  such  cases,  are  exceedingly  uncertain.  The  cir- 
cle NE  (Fig.  84)  represents  the  orbit  of  Neptune  ;  S,  the  sun  ; 
ana  SC,  the  orbit  of  the  comet  of  1849.  The  focus  is  very  close 
to  the  vertex  of  the  ellipse,  represented  by  the  dot. 

Fig.  84. 


364.  IMley's  comet.— This  is  the  only  cornet  of  long  period, 
the  elements  of  whose  orbit  are  all  known.  It  is  a  comet  of 
considerable  splendor,  and  describes  its  orbit  in  75  or  76  years. 
Halley  observed  it  in  1682;  and  finding,  by  computation,  that 
its  path  was  nearly  identical  with  those  of  the  comets  of  1607 
and  1531,  he  conjectured  that  the  three  were  one  and  the  same 
comet,  and  predicted  that  it  would  return  early  in  1759.  It 
did  return  March  12th  of  that  year,  and  again  on  the  16th  of 


188 


COMETS   OF   SHORT   PERIODS. 


November,  1835.  On  its  last  return,  it  reached  the  perihelion 
within  two  days  of  the  calculated  time.  What  renders  such 
agreement  remarkable  is  not  merely  the  great  length  of  the 
periodic  time,  but  the  great  allowance  to  be  made  for  the  dis- 
turbing influence  of  the  principal  planets.  On  its  last  return 
but  one,  the  period  of  this  comet  had  been  diminished  nearly 
two  years  by  the  attractions  of  Jupiter  and  Saturn.  The  aphe- 
lion is  about  600,000,000  miles  beyond  the  orbit  of  Neptune. 
Figure  85  presents  the  form  of  the  orbit  of  Halley's  comet,  and 
its  magnitude  in  comparison  with  the  larger  planetary  orbits. 
The  eccentricity  is  nearly  0.97  ;  hence  the  distance  of  the  focus 
from  the  vertex  is  only  .03  of  the  semi-major  axis.  The  dot  at 
the  left  hand  correctly  represents  the  place  of  the  focus  occu- 
pied by  the  sun. 

Fig.  85. 


is 


r 


jsr 


365.  Comets  of  short  period,  whose  orbits  are  known. — 
There  are  six  of  these,  named  from  the  persons  who  first  pre- 
dicted their  returns.  They  are  telescopic  comets,  and  possess 
no  interest,  therefore,  except  for  the  astronomer.  Their  names, 
periods,  and  extreme  distances  are  given  in  the  following  table. 


COMET. 

Period 
in  years. 

Perihelion 
Distance. 

Aphelion 
Distance. 

Encke's  

31 

32,000,000 

390,000,000 

Biela's  

85,000,000 

570,000,000 

Faye's  

71 

161,000,000 

565,000,000 

Brorsen's 

5* 

62,000,000 

538,000,000 

D'Arrest's  

62 

111,000,000 

546,000,000 

Winnecke's  

5l 

73,000,000 

526,000,000 

189 

It  will  be  perceived  by  the  table,  that  these  six  comets  con- 
siderably resemble  each  other  in  period  and  distance.  They 
are  also  alike  in  being  telescopic,  and  nearly  or  entirely  desti- 
tute of  tails,  and  in  moving  from  west  to  east  at  inclinations  to 
the  ecliptic  not  larger  than  15°  or  20°. 

366.  A.   resisting  medium. — Two   of   the    above   comets, 
Encke's  and  Faye's,  have  given  decided  indications  of  acceler- 
ation in  their  orbits.     This  shows  that  they  meet  with  some 
obstruction,  which  diminishes  their  projectile  force,  in  conse- 
quence of  which  the  centripetal  force  draws  them  into  a  smaller 
orbit,  which  is,  of  course,  described  in  less  time. 

Some  suggest  that  as  the  received  theory  of  light  requires  the 
existence  of  a  medium  throughout  space,  a  substance  of  so  little 
density  as  a  comet  may  possibly  be  obstructed  by  it  sufficiently 
to  render  the  diminution  of  period  perceptible. 

According  to  others,  the  obstruction  may  arise  from  collision 
with  innumerable  small  bodies  which  revolve  about  the  sun. 
The  earth  is  meeting  with  such  bodies  incessantly,  as  is  proved 
by  the  numerous  shooting  stars  which  are  continually  striking 
into  the  atmosphere.  It  is  reasonable  to  suppose  that  other 
bodies,  as  well  as  the  earth,  also  meet  them  ;  and  that  the 
thinnest  and  lightest  bodies,  such  as  the  comets,  should  show 
the  effect  of  such  collisions. 

If  Encke's  and  Faye's  comets,  in  either  of  these  ways,  are 
gradually  diminishing  their  periodic  times,  then  every  other 
comet  must  by  and  by  exhibit  the  same  change ;  and  the  time 
will  come  eventually  when  all  this  class  of  bodies  will,  at  their 
respective  perihelia,  approach  so  near  as  to  fall  upon  the  sun, 
and  be  combined  with  its  substance. 

367.  Division  of  Bields  comet. — One  of  the  most  remark- 
able facts  which  has  occurred  in  the  history  of  comets  was  the 
division  of  Biela's  comet  into  two  distinct  comets.     This  ap- 
pearance was  first  noticed  at  its  return,  in   1846.     The  two 
comets  were  unequal  in  size,  and  the  larger  had  a  short  tail ; 
the  other  was  only  a  little  elongated ;  whereas,  before  the  di- 
vision, the  comet  was  spherical,  without  any  appearance  of  a 
tail.     Since  the  division  took  place,  the  two  bodies  have  moved 


190  REMARKABLE    COMETS. 

in  separate  and  independent  orbits.     Their  distance  apart  in 
1 852  was  about  1 J  millions  of  miles. 

368.  Oilier  remarkable  comets. — A  few  other  comets  are 
here  mentioned,  which,  on  account  of  their  splendor,  or  for 
some  other  reason,  are  regarded  as  objects  of  special  interest. 

369.  The  comet  of  1680.— (Fig.  77).     This  was  a  comet  of 
unusual  brilliancy,  and  appears  to  have  been  the  first  whose 
elements  were  calculated  by  Newton.    At  perihelion,  its  center 
was  only  130,000  miles  from  the  surface  of  the  sun;  so  that,  if 
its  diameter  was  as  large  as  that  of  many  comets,  it  must  have 
come  in  contact  with  it.     Its  velocity  at  perihelion  was  suffi- 
cient to  have  carried  it  round  the  sun  at  that  distance  in  one 
minute. 

370.  The   comet  of  1744.— This  was  the  most  splendid 
comet  of  the  18th  century.      The  remarkable  features  of  it 
were,  its  great  brightness,  and  the  number  of  its  tails.     Its 
light  was  nearly  equal  to  that  of  Yenus,  and  it  was  distinctly 
seen  in  the  daytime,  even  by  the  naked  eye.     After  passing 
the  perihelion,  its  tail  was  spread  into  six  distinct  branches, 
near  40°  in  length,  and  the  extreme  ones  diverging  about  as 
many  degrees  from  each  other. 

371.  The   comet  of  1770.— The  great  interest  wjiich  at- 
taches to  this  comet  arises  from  the  fact  that  it  has  twice  suf- 
fered a  great  change  of  orbit,  in  consequence  of  the  disturbing 
action  of  Jupiter.     It  first  appeared  in  1770,  shining  with  con- 
siderable splendor.      In  1776,  it  again  passed  the  perihelion, 
and  has  never  been  seen  since.      Computations  made  by  La 
Place  and  others  showed  that,  before  its  first  appearance,  it  had 
revolved  in  a  large  orbit,  beyond  our  vision,  and  had  a  period 
of  48  years.     In  1767,  it  came  near  Jupiter,  and  lost  so  much 
of  its  velocity,  that  it  was  drawn  into  a  small  orbit,  wrhoso 
perihelion  was  far  within  the  orbit  of  the  earth.     As  two  of  it? 
revolutions  were  about  equal  to  one  of  Jupiter,  it  was  predicted 
that  it  would  be  again  subjected  to  a  great  disturbance  at  its 
aphelion.     This  actually  took  place  in  1779,  since  which  time 


REMARKABLE    COMETS. 


191 


it  lias  never  been  seen.  Its  present  period  is  calculated  to 
be  about  20  years,  AB  (Fig.  86)  is  a  part  of  Jupiter's  orbit ; 
E  is  the  earth's  orbit ;  CD  is  the  path 
of  the  comet  before  1770.  Near  D. 
it  was  so  retarded  by  Jupiter,  that  the 
sun  drew  it  into  the  small  orbit  DFH,  B 
which  it  described  twice,  when,  near 
D,  it  was  again  powerfully  affected 
by  Jupiter,  and  received  a  great  accel- 
eration, which  caused  it  to  pass  out 
once  more  into  a  large  orbit,  DK. 


372.  The    comet    of   1843.— The 
brightness  of  this  comet  was  so  great, 
that    it    was    seen   during  the   day. 
Its  perihelion  distance  was  less  than 
100,000  miles,  and  its  exterior  parts 
were  probably  in  actual  contact  with 
the  sun.     No  other  comet  has  been 

known  to  approach  so  near.  The  tail  spanned  70°  of  the 
sky,  and  was  unusually  straight  and  slender,  as  exhibited  in 
PL  I. 

373.  The   comet  of  1858.— This  is   also   called   Donati's 
comet,  having  been  discovered  by  Donati,  of  Florence.     It  was 
remarkable   for   the   series   of   envelopes   formed   successively 
about  the  nucleus,  as  it  approached  the  perihelion.     The  ap- 
pearance of  the  head  is  shown  in  PI.  II.,  Fig.  1,  and  the  entire 
comet  in  Fig.  2.     Its   period   is  computed  to  be  about  two 
thousand  years. 

374.  The  comet  of  1801. — This  comet  came  so  near  the 
earth,  that  it  is  believed  a  part  of  the  tail  swept  across  it.     But 
it  is  not  certain  that  any  visible  effect  was  produced.     The 
apparent  length  of  its  tail,  at  one  time,  was  106°.     Fig.  87 
shows  its  form  at  that  time. 


375.  Effects  of  collision  between  a  planet  and  a  comet. — 
Whether  a  direct  collision  between  the  earth  and  the  nucleus 


192 


SHOOTING   STARS. 


Fig.  87. 


of  a  comet  would  produce  serious  ef- 
fects, it  is  impossible  to  know,  because 
so  little  is  understood  respecting  the 
density  of  the  nucleus.  But  the  coma 
and  tail  consist  of  matter  thousands  of 
times  more  rarefied  than  the  earth's 
atmosphere,  and  would  probably  fail 
to  penetrate  it  at  all.  The  earth  is 
thought  to  have  passed  through  a 
comet's  tail,  at  least  in  one  instance, 
but  without  producing  any  perceptible 
effect. 

376.  Shooting  stars. — This   is   the 
popular  name  given   to    those   bodies 
which  appear  like  stars  or  planets  mov- 
ing across  some  part  of  the  sky,  and 
then  vanishing.     They  are  equally  well 
known  by  the  name  of  meteors.     They 
may  be   seen  in   any  clear  night,  by 
watching  an  hour  or  two,  especially  if 
the  moon  is  not  shining. 

377.  Height  and  velocity. — By  means 
of  concerted  observations,  made  at  sta- 
tions quite  distant  from  each  other,  the 
angle  can  be  measured,  which   is  in- 
cluded by  lines  drawn  from  a  meteor  to  the  stations,  both  at 
the  beginning  and  end  of  its  motion,  and  thus  its  distance 
and  velocity  can  be  measured.     The   heights  of  meteors  are 
thus  found  to  be  generally  about  50  miles,  and  their  velocities 
20  or  30  miles  per  second.     Coming  into  the  air  with  such 
great  velocity,  they  are  almost  instantly  set  on  fire,  and  their 
substance  becomes  incorporated  with  the  atmosphere. 

378.  Gaseous  meteors. — If  the  ordinary  meteors  were  more 
dense  than  a  gas,  they  would  hardly  lose  all  their  motion,  as 
they  do,  before  reaching  the  earth.     The  most  interesting  facts 
relating  to  this  class  of  bodies  are  the  following : 


SHOOTING   STARS.  193 

1.  They  often  occur  in  showers — that  is,  thousands  and  hun- 
dreds of  thousands  of  them  are  seen  in  a  single  night. 

2.  These  showers  have  periodical  returns. 

3.  The  meteors  of  a  shower  come  into  the  atmosphere  in  a 
given  direction,  or,  in  other  words,  in  parallel  lines.     The  op- 
tical effect  is,  that  they  appear  to  describe  arcs  of  great  circles, 
having  a  common  place  of  intersection. 

379.  Dates  of  meteoric  showers. — The   most  remarkable 
meteoric  shower  of  the  present  century  was  November  12-13, 
1833.     Not  less  than  200,000  meteors  were  seen  during  the 
night  at  any  one  station.     A  like  shower  occurred  at  the  same 
time  in  1799.     And  generally,  there  are  more  meteors  about 
the  12th  of  November,  than  at  any  other  time  of  the  year. 

Other  dates  at  which  meteors  are  unusually  abundant  are 
April  21st,  August  9th,  and  December  7th. 

380.  Origin  of  the  gaseous  meteors. — The  known  motion  of 
the  earth,  and  the  observed  velocity  and  direction  of  this  class 
of  bodies,  lead  to  a  knowledge  of  their  heliocentric  motions. 
It  is  found  in  this  way  that  they  describe  ellipses  about  the 
sun,  and  are  therefore  to  be  regarded   as  minute   cometary 
bodies.     Those  which  come  in  showers  seem  to  belong  to  ex- 
tensive groups,  which  revolve  about  the  sun  in  zones  or  rings. 
There  appear  to  be  three  or  four  of  these  zones,  whose  planes 
are  situated  at  different  obliquities  to  the  ecliptic,  and  across 
which  the  earth  passes  once  a  year.     When  the  earth  traverses 
a  more  crowded  portion  of  such  a  ring  of  meteors,  the  phenom- 
enon of  a  meteoric  shower  occurs. 

381.  Solid  meteors. — There  is   another   class   of  meteoric 
bodies,  which  afford  indubitable  evidence  of  being  solid.     Like 
the  gaseous  meteors,  they  plunge  into  the  atmosphere  with 
great  velocity,  and  are  inflamed  by  the  violent  attrition.     Be- 
fore reaching  the  earth  they  usually  explode,  and  scatter  their 
fragments.     Some  of  them,  ho\vever,  appear  to  lose  only  small 
portions  of  their  mass  by  explosion,  and  pass  o-n  in  their  orbits 
around  the  sun — greatly  disturbed,  of  course,  by  the  earth's  at- 
traction. 

18 


194  THE    STELLAR    UNIVERSE. 

382.  Aerolites. — This  is  the  name  usually  given  to  the  frag- 
ments thrown  down  by  solid  meteors ;  though  in  rare  instances, 
an  aerolite  obviously  constitutes  the  entire  meteor  itself.  Aer- 
olites consist  of  iron,  silex,  and  a  few  other  materials,  which 
are  all  known  among  terrestrial  substances.  But  they  are 
always  distinguishable  from  terrestrial  bodies  by  their  peculiar 
structure.  Since  the  great  velocities  of  meteors,  solid  as  well 
as  gaseous,  have  become  known,  the  former  theories  as  to  the 
origin  of  meteoric  stones,  or  aerolites,  have  been  abandoned. 
Such  velocities,  if  they  could  be  generated  at  all  on  the  earth, 
could  never  exist  in  horizontal  or  downward  directions.  Both 
solid  and  gaseous  meteors  are  therefore  considered  as  describ- 
ing orbits  about  the  sun.  The  interplanetary  spaces,  which 
have  been  generally  reckoned  as  vacant,  may  perhaps  be  to  a 
great  extent  occupied  by  innumerable  bodies,  of  a  grade  far 
below  that  of  comets  and  planetoids. 


CHAPTEE  XIX. 

THE  FIXED  STARS. — THEIR  CLASSIFICATIONS. — THEIR  DIS- 
TANCES AND  MOTIONS. — DOUBLE  STARS,  CLUSTERS,  AND 
NEBULAE. — THE  NEBULAR  HYPOTHESIS. 

383.  The  stellar  universe. — The   bodies  described  in  the 
foregoing  chapters  all  belong  to  the  solar  system.    If  our  inves- 
tigations are  extended  outside  of  this  system,  we  find  that  there 
are  other  systems,  greater  or  less  than  this,  unlimited  in  num- 
ber, and  separated  from  the  solar  system  and  from  each  other 
by  solitudes  so  vast,  that  each  system  is  only  a  point  in  com- 
parison with  the  distances  between  them.     The  central  sun  in 
each  of  these  countless  systems  is  a  fixed  star. 

The  word  "  universe"  is  employed  to  express  the  sum  total 
of  all  these  systems,  the  number  of  which,  and  the  extent  of 
space  occupied  by  them,  are  utterly  beyond  the  reach  of  human 
comprehension. 

384.  The  fixed  stars,   and  their  magnitudes.—  -The  fixed 


MAGNITUDES    OF    STAES.  195 

stars  are  so  called,  because,  to  common  observation,  they  always 
maintain  the  same  situations  with  respect  to  each  other.  All 
the  thousands  of  bright  points  ordinarily  seen  in  the  sky  by 
night  are  fixed  stars,  with  the  exception  of  two  or  three,  possi- 
bly four,  which  are  planets. 

The  fixed  stars  are  classified  according  to  magnitudes,  though 
the  word,  when  thus  used,  signifies  only  degrees  of  brightness. 
The  stars  which  can  be  seen  by  the  naked  eye,  in  the  most 
favorable  circumstances,  are  divided  into  six  magnitudes. 
Those  which  can  be  seen  only  by  the  aid  of  the  telescope, 
called  telescopic  stars,  are  arranged  into  several  more ;  so  that 
all  the  magnitudes  are  16  or  18. 

Stars  of  the  same  magnitude  are  not  equally  bright ;  for 
there  is  a  continual  gradation  in  respect  to  brightness ;  so  that, 
if  the  intensity  were  accurately  measured,  probably  the  light 
of  but  very  few  would  be  found  exactly  equal. 

•Stars  of  the  first  magnitude  are  fewest  in  number,  and,  gen- 
erally, the  smaller  the  magnitude,  the  larger  the  number  of 
stars  included  under  it.  The  limits  of  the  successive  magni- 
tudes differ  somewhat,  according  to  different  astronomers ;  but 
the  following  round  numbers  do  not  vary  widely  from  any  of 
them. 


1st  magnitude    .     .     20 
2d          "  .     .     40 

3d          "  .  HO 


4th  magnitude  .  .300 
5th  "  .  .  950 
6th  "  .  4450 


In  all,  near  6,000,  visible  to  the  naked  eye.  The  numbers  of 
the  telescopic  stars  increase  at  so  rapid  a  rate,  that  they  have 
to  be  reckoned  by  millions. 

385.  Cause  of  unequal  brightness. — We  might  suppose 
either  that  the  stars  are  themselves  unequal  in  respect  to  the 
quantity  of  light  which  they  emit,  or  that  they  appear  un- 
equally bright  on  account  of  their  different  distances.  It  is 
undoubtedly  true  that  there  is  some  diversity  in  the  bodies 
themselves ;  and  yet,  the  rapid  increase  of  numbers  as  the  mag- 
nitudes are  less,  indicates  that  difference  of  distance  is  the  chief 
cause  of  inequality  in  brightness.  If  there  is  any  approach  to 


196  CONSTELLATIONS. 

a  uniform  distribution  of  the  stars  in  space,  those  which  are 
nearest  should  be  fewest  in  number,  and  should,  in  general, 
appear  brightest. 

386.  Constellations. — The  fixed  stars  are  also  classed  topo- 
graphically in  constellations.  This  division  is  very  ancient ; 
and  some  of  the  constellations  are  mentioned  by  the  earliest 
writers.  The  names  given  to  them  are  those  of  the  animals, 
heroes,  and  other  objects  of  pagan  mythology. 

Constellations  of  the  zodiac. 

Aries.  Libra. 

Taurus.  Scorpio. 

Gemini.  Sagittarius. 

Cancer.  Capricornus. 

Leo.  Aquarius. 

Yirgo.  Pisces. 

Constellations  north  of  the  zodiac. 

Ursa  Major.  Auriga.  Cygnus. 

Ursa  Minor.  Leo  Minor.  Yulpecula. 

Draco.  Canes  Yenatici.  Aquila. 

Cepheus.  Coma  Berenices.  Antinous. 

Cassiopeia.  Bootes.  Delphinus. 

Camelopardalus.        Corona  Borealis.  Pegasus. 

Andromeda.  Hercules.  Ophiuchus. 

Perseus.  Lyra. 

Constellations  south  of  the  zodiac. 

Cetus.  Monoceros.  Hydra. 

Orion.  Canis  Major.  Crater. 

Lepus.  Canis  Minor.  Corvus. 

Centaurus.  Crux.  Eridanus. 

Lupus.  Argo  Kavis. 

The  foregoing  are  the  principal  constellations ;  but  several 
more,  mostly  small  ones,  may  be  found  on  globes  and  charts. 


ANNUAL  PARALLAX.  197 

Within  each  constellation,  the  brightest  stars  are  designated 
by  the  letters  of  the  Greek  alphabet  in  the  order  of  brightness. 
Thus,  a  Lyrse,  is  the  brightest  star  in  Lyra ;  (3  Scorpionis,  the 
brightest  but  one  in  Scorpio,  etc.  After  the  Greek  letters  are 
all  used,  Roman  letters,  and  then  numerals,  are  employed.  In 
some  cases,  the  order  of  brightness  does  not  accord  with  the 
order  of  the  alphabet.  This  may  result  from  a  change  of 
brightness,  which  has  taken  place  since  the  stars  were  first 
named.  When  a  capital  letter  follows  a  number,  there  is  ref- 
erence to  the  catalogue  of  some  astronomer.  Thus,  81H  is  the 
star  84  of  a  certain  constellation  in  Herschel's  catalogue. 

A  few  conspicuous  stars  are  still  known  by  individual  names 
given  to  them  in  ancient  times ;  as  Arcturus,  Antares,  Sirius, 
Yega,  etc. 

The  first  catalogue  of  stars  was  made  by  Hipparchus,  before 
the  time  of  Christ,  and  contained  1,022  of  the  most  conspicuous 
stars.  Catalogues  of  the  present  day  contain  hundreds  of  thou- 
sands of  stars,  whose  right  ascensions  and  declinations  are  given 
for  a  certain  date. 


387.  Effect  of  telescopic  power  on  fixed  stars. — One  indica- 
tion of  the  vast  distance  of  the  fixed  stars  is,  that  no  power  of  a 
telescope  has  sensibly  magnified  them.     Even  under  a  power 
which  increases  the  diameter  of  a  body  5,000  times,  they  appear 
no  larger  than  to  the  naked  eye.     It  is  inferred  that  they  fill  an 
angle  so  small,  that  5,000  times  that  angle  is  still  too  minute  to 
be  perceived.     Any  appearance  of  disk  which  a  star  presents, 
either  with  a  telescope  or  without,  is  the  effect  of  the  light 
upon  the  retina  of  the  eye.     It  is  called  a  spurious  disk,  since 
an  increase  of  magnifying  power  causes  no  increase  of  its  di- 
ameter. 

388.  Annual  parallax. — Another  proof  that  the  fixed  stars 
are  at  an  immense  distance  from  us,  is  the  fact  that  while  we 
shift  our  position  every  six  months  from  one  side  of  the  earth's 
orbit  to  the  opposite,  a  distance  of  190,000,000  miles,  there  is 
no  perceptible  change  in  the  relation  of  the  stars  to  each  other. 
It  is  only  after  long-continued  and  most  accurate  observation, 


198  ANNUAL  PARALLAX. 

that  a  few  stars  have  been  discovered  to  suffer  an  annual 
change  of  position,  which  is  clearly  of  the  nature  of  paral- 
lax. 

The  annual  parallax  of  a  star  is  the  angle,  at  the  star,  sub- 
tended by  the  radius  of  the  earth's  orbit.  As  this  angle  is  in 
almost  all  cases  too  small  to  be  detected,  it  shows  that  the 
earth's  orbit,  seen  from  the  distance  of  the  stars,  appears  as  a 
mere  point. 

389.  The  parallactic  path  of  a  star. — If  the  annual  paral- 
lax of  a  star  is  in  any  case  perceptible,  its  apparent  movement 
during  the  year  depends  entirely  on  its  situation  in  relation  to 
the  ecliptic. 

A  star  in  the  plane  of  the  ecliptic  will  appear  to  oscillate 
back  and  forth  in  a  straight  line  once  in  a  year.  It  will  appear 
stationary  at  the  two  opposite  seasons,  when  the  earth  is  going 
toward  it,  and  from  it ;  and  if  we  imagine  a  diameter  of  the 
earth's  orbit  joining  these  two  positions,  the  star  will  seem  to 
describe  a  straight  line  parallel  to  that  diameter,  its  motion 
during  each  half-year  being  opposite  to  the  general  direction  of 
the  earth's  motion. 

But  if  a  star  at  the  pole  of  the  ecliptic  should  exhibit  any 
parallax,  its  apparent  motion  would  be  in  an  orbit  parallel  to 
the  earth's  orbit,  and  similar  to  it :  it  may  be  regarded,  there- 
fore, as  a  circle  described  about  the  point  in  which  the  star 
would  be  seen  from  the  sun.  Moreover,  the  star's  apparent 
place,  and  the  earth's  real  place  in  their  respective  orbits  would 
be  diametrically  opposite. 

At  a  point  between  the  plane  of  the  ecliptic  and  its  pole, 
the  parallactic  orbit  wTould  be  an  ellipse,  the  ratio  of  whose 
axes  would  depend  on  the  latitude  of  the  star. 

390.  Discovery  of  annual  parallax. — It  is  justly  reckoned 
among  the  greatest  achievements  in  practical  astronomy,  that 
the  annual  parallax  has,  in  a  few  cases,  not  only  been  clearly 
detected   as   existing,   but  has    been   satisfactorily  measured, 
though  it  is  never  so  great  as  V . 

The  parallax  of  a  Centauri  is  0".91 ;  that  of  61  Cygni,  0".?5; 
of  a  Lyrse,  0".26  ;  of  Sirius,  0".23.  A  few  others  have  been 


DISTANCES    OF   STARS.  199 

obtained,   which   are  still   smaller,   and  therefore  less    relia- 
ble. 

The  parallax  of  a  star  is  most  satisfactorily  determined,  when 
it  is  in  the  same  telescopic  field  with  other  stars.  For  then 
the  distances  between  the  stars  may  be  measured  with  great 
precision  by  a  micrometer,  and  all  errors  arising  from  aberra- 
tion, refraction,  and  instrumental  disturbance  are  wholly 
avoided,  because  all  the  stars  in  the  same  field  are  affected 
alike  by  these  causes  of  displacement.  Parallax  is  the  only  cir- 
cumstance which  can  produce  an  annual  change  in  their  rela- 
tive positions.  The  star  61  Cygni  is,  in  this  respect,  very 
favorably  situated,  and  its  parallax  is  thought  to  be  quite 
accurately  determined. 

391.  Distances  of  those  stars  whose  parallax  is  known. — 
If  a  triangle  is  formed  by  the  lines  joining  the  sun,  earth,  and 
star,  and  the  angle  at  the  sun  be  a  right  angle,  we  have  the 
proportion 

Sin  an.  par. :  rad  : :  95,134,000  miles  :  dist.  of  the  star. 
This  gives  the  distance  of  «  Centauri,  the  nearest  star, 
22,000,000,000,000  miles,  nearly.  Light,  moving  at  the  rate 
of  192,500  miles  per  second,  would  require  about  3.5  years  to 
come  from  that  star  to  us;  9.3  years  from  61  Cygni ;  12.4 
years  from  a  Lyrse ;  and  14.1  years  from  Sirius.  And  if  we 
reckon  the  parallax  of  the  pole-star  at  0".07,  as  it  has  been 
computed  to  be,  it  requires  48  years  for  its  light  to  reach  us. 

In  order  to  compare  these  amazing  distances  with  the  dimen- 
sions of  the  solar  system,  we  may  use  with  advantage  the  dia- 
gram described  in  the  note,  Art.  263.  The  distance  from  the 
sun  to  Neptune  being  represented  by  30  feet,  the  distance  of 
the  nearest  star,  «,  Centauri,  must  be  represented  by  40  miles, 
and  that  of  61  Cygni  by  110  miles,  etc.  Thus  isolated  are  the 
systems  of  the  universe  from  eacli  other. 

As  to  all  other  stars  besides  those  above  named,  it  is  only 
known  that  they  are  still  more  distant.  There  is  no  improb- 
ability that,  from  the  remotest  telescopic  stars  yet  seen,  light 
may  occupy  thousands  of  years  in  coming  to  us.  Therefore, 
we  see  all  the  stars  as  they  were  years  ago ;  perhaps  not  as 
they  are  now.  And  if  at  any  time  a  change  has  been  detected 


STARS    ARE    SUNS. 

in  the  aspect  or  place  of  a  star,  that  change  occurred,  not  when 
it  was  seen,  but  10,  100,  or  1,000  years  before,  according  to  its 
distance. 

392.  Nature  of  the  fixed  stars. — The  stars  are  situated  at 
such  vast  distances  from  the  solar  system,  that  if  they  merely 
reflected  the  light  of  the  sun,  they  would  be  invisible.  In 
order  to  exhibit  such  brightness  as  they  do,  they  must  not  only 
shed  light,  but  a  very  intense  light  of  their  own.  They  can 
not  be  compared  with  any  one  of  the  bodies  in  the  solar  sys- 
tem, except  the  sun  iteelf.  All  the  fixed  stars,  therefore,  are  to 
be  considered  as  suns,  and  probably  the  centers  of  systems  re- 
sembling the  solar  system.  It  is  ascertained,  respecting  some 
of  those  stars  whose  distance  is  known,  that  they  shed  more 
light  than  the  sun.  For  example,  a  Centauri  has  been  found 
to  shed  near  four  times  as  much  light  as  the  sun.  For  the 
light  of  the  sun  at  the  earth  is  about  500,000  times  as  great  as 
the  light  of  the  full  moon.  And  the  light  of  the  full  moon  was 
found  by  Sir  John  Herschel's  observations  to  equal  27,000 
times  that  of  a  Centauri.  Therefore,  the  light  of  the  sun  at 
the  earth  is  (500,000  x  27,000)  13,500,000,000  times  that  of 
a  Centauri  at  the  earth.  But  that  star  is  230,000  times  as  far 
off  as  the  sun.  And  since  the  quantity  of  light  received  from 
a  luminous  body  varies  inversely  as  the  square  of  the  distance, 
if  «  Centauri  were  brought  as  near  to  us  as  the  sun,  its  light 
would  be  52,900,000,000  (=  230,000)2  times  as  great  as  it  is  at 
present,  or  nearly  four  times  as  great  as  the  light  of  the  sun. 

In  a  similar  manner,  Sirius,  the  brightest,  but  not  the 
nearest  fixed  star,  is  found  to  shed  100  times  as  much  light  as 
the  sun. 

On  the  other  hand,  if  the  sun  were  removed  from  us  to  the 
nearest  fixed  star,  its  apparent  diameter  would  be  only  T  J^",  and, 
therefore,  would  be  a  star  having  no  sensible  magnitude,  and 
having  only  T^o  of  the  brightness  of  Sirius. 

393.  Proper  motion  of  the  stars. — There  is  increasing  evi- 
dence that  there  is  among  the  stars  a  parallactic  motion  of  a 
higher  order  than  the  annual  parallax  already  noticed.  The 
entire  solar  system  appears  to  be  moving  toward  a  certain 


DOUBLE    STARS.  201 

point  in  the  constellation  Hercules,  whose  right  ascension  is 
260°,  and  its  declination  55°  north.  This  motion  of  the  system 
is  inferred  from  what  is  termed  th&  proper  motion  of  the  stars. 
Since  the  time  of  Hipparchus  (130  B.  0.),  Sirius,  Arcturus, 
and  Aldebaran  have  changed  their  position  southward  more 
than  half  a  degree.  The  star  61  Cygni  moves  5"  each  year, 
\L  Cassiopeia  4",  and  e  Indi  8" ;  and  a  large  number  of  other 
stars  have  a  small  progressive  motion.  The  general  effect  of  a 
motion  of  our  own  system  would  be  to  cause  a  minute  ap- 
parent separation  of  the  stars  in  the  region  toward  which  we 
are  moving,  and  a  crowding  together  of  the  stars  in  the  region 
from  which  we  move.  From  a  comparison  of  the  proper  mo- 
tions of  several  hundreds  of  stars,  a  motion  of  the  solar  system 
in  the  direction  named  above  has  been  deduced.  And  the  rate 
of  that  motion  has  been  estimated  to  be  about  154.000,000 
miles  per  year,  which  is  only  one-fourth  the  earth's  velocity  in 
its  orbit. 

If  the  motion  is  really  perceptible,  it  is  probable  that  a 
change  of  direction  will,  after  a  few  centuries,  manifest  itself, 
from  which  something  may  be  inferred  as  to  the  position  and 
magnitude  of  the  orbit  which  the  sun  describes. 

Some  of  the  stars  have  a  proper  motion,  which  can  not  be  ex- 
plained by  the  supposed  motion  of  the  solar  system.  In  those 
cases,  it  must  be  concluded  that  they  are  themselves  describing 
vast  system- orbits  about  some  distant  center. 

394.  Double  stars. — It  is  discovered  in  a  great  number  of 
instances  that  a  fixed  star,  when  examined  by  the  telescope, 
really  consists  of  two  stars,  very  close  to  each  other.  If  the 
distance  between  them  does  not  exceed  32",  such  stars  are 
called  double  stars.  Their  distance  apart  is  often  less  than  1", 
and  some  are  so  close,  that  the  highest  power  of  the  telescope 
and  the  most  acute  vision  are  requisite  to  separate  them. 
Hence,  certain  double  stars  are  habitually  used  as  tests  of  the 
excellence  of  an  instrument. 

When  Sir  "William  Herschel  first  began  his  observations  on 
this  class  of  objects,  in  1780,  he  knew  of  only  four  j  but  he  ex- 
tended the  list  to  500  himself,  and  the  number  now  known  ex- 
ceeds 6,000. 


202  DOUBLE    STAES. 

395.  Relative  intensity  and  color. — In  comparatively  few 
instances  are  the  two  stars  equally  bright.     They  sometimes 
differ  so  little  as  to  fall  within  the  limits,  of  the  same  magni- 
tude; but  generally  they  are  of  different  magnitudes.     Thus, 
the  component  stars  of  y  Leonis  are  of  the  2d  and  4th  magni- 
tudes ;  of  T]  Lyrse,  4th  and  8th ;  and  of  the  pole-star,  2d  and 
9th.     Figures  1,  2,  3,  and  4,  in  PL  III.,  present  the  telescopic 
appearance  of  the  double  stars  there  named.     In  4,  they  are  so 
close  as  to  appear  like  a  single  star,  of  tapering  form. 

A  fact  of  great  interest,  in  relation  to  double  stars  is,  that 
they  often  differ  in  color.  Sometimes  these  colors  are  comple- 
mentary;  that  is,  they  are  such  as  would  compose  white  light, 
if  mingled  together.  In  such  cases,  if  the  stars  differ  much  in 
magnitude,  the  appearance  of  color  in  the  fainter  star  may  be 
only  an  illusion.  But  this  can  not  be  true  when  the  colors  are 
not  complementary.  The  components  of  y  Andromedse  are 
orange  and  green;  of  %  Bootis,  white  and  violet;  of  a  Herculis. 
yellow  and  blue  ;  and  of  j3  Scorpionis,  white-and  blue. 

Single  stars  are  frequently  of  a  deep  red  color ;  but  a  decided 
case  of  green  or  blue  is  never  met  with,  except  in  a  component 
of  a  double  star. 

396.  Two  ways  in  which  stars  might  appear  double. — The 
two  stars  which  compose  a  double  star  may  be  supposed  either 
to  be  really  near  each  other,  or  only  to  appear  near  together, 
because  they  fall  almost  into  the  same  line  of  vision,  while  one 
is  actually  at  an  immense  distance  beyond  the  other.     In  the 
latter  case,  the  stars  are  said  to  be  optically  double.    When  Sir 
William  Herschel  commenced  examining  double  stars,  he  very 
naturally  supposed  that,  in  the  very  few  cases  known,  one  star 
happened   thus  to  be  nearly  in  the  same  visual  line  with  the 
other;  and  he  began  the  work  of  observing  them,  with  the  ex- 
pectation of  detecting  annual  parallax  in  objects  so  favorably 
situated.     For.  if  the  nearer  star  is  perceptibly  affected  by  par- 
allax, it  would  exhibit  an  annual  motion  relatively  to  the  more 
distant  star,  in  a  manner  not  to  be  mistaken. 

397.  Binary  stars. — It  soon  became  evident,  however,  that 
double  stars  are  too  numerous  to  allow  the  supposition  that 


ORBITS   OF  BINARY    STARS.  203 

their  apparent  proximity  is  only  casual.  It  was  calculated 
that  the  chance,  that  of  all  the  stars  visible  to  the  naked  eye, 
two  would  accidentally  appear  within  4"  of  each  other,  was 
only  1  in  9,000 ;  whereas  one  hundred  such  cases  were  already 
known. 

But  another  most  interesting  discovery  was  presently  made ; 
namely,  that  some  of  the  double  stars  exhibit  motions  which 
indicate  a  revolution  of  one  around  the  other — or,  rather,  of  the 
two  around  a  common  center,  and  in  periods  of  various  lengths, 
having  no  connection  whatever  with  the  earth's  annual  motion. 
Such  motion  can  not  be  parallactic ;  it  must  be  real ;  and  such 
stars  are  not  optically,  but  physically  double.  They  are  called 
binary  stars,  and  are  to  be  regarded  as  the  centers  of  double 
stellar  systems. 

398.  Gravitation  outside  of  the  solar  system. — The  binary 
stars  afford  evidence  that  the  same  law  of  attraction  which  pre- 
vails within  the  boundary  of  the  solar  system  prevails  also  at 
immeasurable  distances  beyond  it.     In  the  case  of  every  binary 
star  which  has  yet  completed  the  whole,  or  any  considerable 
part  of  its  revolution,  since  its  discovery,  it  is  found  that  the 
path  of  one  component  star  is  an  ellipse,  while  the  other  occu- 
pies one  of  the  foci  within  it.     Hence,  the  law  of  attraction  is, 
gravity  varies  inversely  as  the  square  of  the  distance,  just  as 
within  the  solar  system.     Though  the  relative  motion  may  be 
represented  by  considering  either  star  as  occupying  the  focus, 
and  the  other  star  as  revolving  about  it,  yet  the  true  focus  is 
the  center  of  gravity  between  them,  while  each  describes  its 
orbit  about  that  center. 

399.  The  real  and  the  apparent  orbit. — It  is  not  to  be  as- 
sumed that  the  plane  of  a  stellar  orbit  is  perpendicular  to  our 
line  of  vision.     But  if  it  is  oblique,  although  it  is  always  pro- 
jected on  the  sky  as  an  ellipse,  yet  the  apparent  eccentricity 
may  differ  in  any  degree  from  the  real  eccentricity,  and  the 
central  star  will  probably  appear  out  of  the  focus  of  the  ap- 
parent orbit.     The  true  orbit,  however,  can  be  readily  deduced 
from  the  apparent  one,  by  means  of  the  position  of  the  central 
star.    If  the  plane  of  revolution  of  a  binary  star  were  coinei- 


204 


PERIODS    OF    BINARY  STARS. 


Fig.  88. 


dent  with  our  line  of  vision,  one  star  would  appear  to  oscillate 
in  a  straight  line  across  the  other. 

The  ellipse,  BCD  (Fig.  88),  represents  the  apparent  orbit  of 
£  Ursse  Majoris,  the  central  star 
being  at  A.     The  real  orbit,  of 
which  A  is  the  focus,  is  BDF. 

The  apparent  orbit  of  a  Centauri 
is  still  more  eccentric  (Fig.  89), 
compared  with  the  real  one,  be- 
cause more  oblique  to  the  line  of 
vision.  It  has  not  yet  described 
quite  half  its  orbit,  since  it  began 
to  be  observed. 

At  the  bottom  of  PI.  III.  are 
shown  the  relative  positions  and 
distances  of  y  Yirginis  from  1 837 
to  1860,  and  the  form  of  the  apparent  orbit.  The  real  orbit 
is  even  more  eccentric,  the  major  axis  being  somewhat  fore- 
shortened by  obliquity. 

Fig.  89. 


4OO.  Periods  of  binary  stars. — The  shortest  period  known 
is  that  of  £  Herculis,  about  31  years.  The  period  of  r\  Coronae 
is  43  years :  that  of  £  Ursae  Majoris  (Fig.  88)  is  58  years. 
These,  and  a  few  others  of  short  period,  have  completed  their 


PERIODS  OF  BINARY  STARS.  205 

revolutions  once  or  twice  since  they  were  discovered.  The 
orbits  of  such  are  quite  accurately  determined.  The  period  of 
a  Centauri  (Fig.  89)  has  not  yet  made  a  revolution  since  its 
discovery ;  its  period  is  calculated  to  be  77  years.  A  large 
number  of  binary  stars,  whose  periods  are  computed  to  be  some 
hundreds  or  thousands  of  years,  have  been  observed  as  yet  only 
through  a  short  arc;  hence  their  periodic  times,  and  the  forms 
of  their  orbits,  are  quite  uncertain. 

401.  Dimensions  of  stellar  orbits. — There  are  two  binary 
stars  whose  parallax  has  been  so  satisfactorily  measured,  that 
their   distances  from  us  may  be  considered  as  well  known ; 
these  are  a  Centauri  and  61  Cygni.     Hence,  by  the  angular 
length  of  the  semi-major  axis  of  their  orbits,  we  may  find  the 
mean  radius  vector  of  each.    The  major  axis  of  the  orbit  of 
a  Centauri  is  about  30",  and  its  distance  from  the  earth  is 
22,000,000,000,000  miles. 

/.  rad  :  sin  15"  : :  22,000,000,000,000  :  1,510,000,000  miles; 
which  is  equalto  about  16  times  the  earth's  distance  from  the 
sun.  The  distance  between  the  components  of  61  Cygni  is 
about  4,200,000,000  miles. 

402.  Masses  of  the  Unary  stars. — For  those  binary  stars 
whose  periods  and  distances  apart  are  known,  the  mass  of  the 

D3 

central  star  can  be  computed.  For  M  QO  ^-2 ;  hence,  for  a  Cen- 
tauri (the  earth's  distance  from  the  sun,  and  its  period  being 

163 
called  1),  M  =  -==-^  =  0.69.    That  is,  the  mass  of  one  component 

of  a  Centauri  is  about  T7o  of  the  mass  of  the  sun.  So,  for  61 
Cygni,  whose  period  is  computed  to  be  540  years,  and  the 
distance  of  the  two  components  44  times  the  radius  of  the 
earth's  orbit,  the  mass  of  the  central  star  is  0.3  of  the  mass  of 
the  sun. 

403.  Triple  and  quadruple  stars. — There  are  a  few  in- 
stances of  three  or  four  stars,  which  are  known  to  be  physical  ly 
connected,  and  to   constitute  a  system.     Figs.  5,  6,  PL  III. 
present  the  appearance  of  11  Monocerotis  and  £  Cancri.     In 


206  PEEIODIC   STAHS. 

the  latter,  the  two  close  components  revolve  in  59  years,  and 
the  distant  one  more  slowly.  The  faint  star  e  Lyrge  is  quadru- 
ple, consisting  of  two  very  close  double  stars.  They  give  evi- 
dence of  belonging  to  one  system,  but  their  revolutions  are  ex- 
ceedingly slow. 

4O4.  Periodic  and  temporary  stars. — There  are  among  the 
fixed  stars  several  instances  in  which  there  appear  to  be  revolu- 
tions of  another  sort,  the  nature  of  which  is  not  understood. 
Stars  which  exhibit  these  changes  are  called  periodic  sta,rs.  A 
remarkable  example  occurs  in  the  star  o  Ceti.  It  passes 
through  its  changes  of  brightness  in  about  11  months.  When 
brightest,  it  is  of  the  2d  magnitude,  and  remains  so  for  two 
weeks.  It  then  diminishes  during  3  months  to  the  10th  mag- 
nitude, remains  thus  5  months,  and  increases  again  during  3 
months  to  its  maximum  of  brightness. 

Algol  ((3  Persei)  has  a  very  short  period,  occupying  only  2d. 
20h.  48m.  Its  changes  succeed  each  other  with  great  regular- 
ity, thus : 

During  2d.  14h.    Om.  it  remains  of  the  2d  magnitude. 
"         Od.    3h.  24m.  diminishes  from  2d  to  4th. 
"         Od.    3h.  24m.  increases  from  4th  to  2d. 


2d.  20h.  48m.  whole  period. 

Some  of  this  class  of  stars  have  periods  of  only  a  few  days, 
while  in  others  the  changes  go  on  very  slowly,  and  appear  to 
require  several  years.  The  periods  of  some  are  quite  uniform, 
arid  of  others  irregular.  As  accurate  observations  are  mul- 
tiplied, the  number  of  known  periodic  stars  is  constantly  in- 
creasing. 

To  this  class  probably  belong  those  stars  which  are  called 
temporary  stars.  That  of  1572  is  celebrated.  It  appeared  so 
suddenly,  and  of  such  brilliancy,  as  to  attract  the  attention  of 
common  people,  and  rapidly  increased,  till  in  a  few  weeks  it 
surpassed  Jupiter  in  brightness.  It  then  faded  slowly,  and 
after  about  1^  years  entirely  disappeared.  Several  other  cases 
less  marked  than  this  are  on  record.  And  the  earlier  cata- 
logues contain  numerous  stars  which  are  not  to  be  found  at  the 
present  day.  Undoubtedly  some  of  these  records  are  mistakes. 


NEBULA.  207 

In  two  or  three  instances,  it  is  known  that  the  bodies  were 
planets,  not  fixed  stars.  But  in  the  course  of  coming  centnries, 
some  of  the  temporary  stars  may  again  become  visible,  and 
thenceforward  be  recognized  as  periodic  stars. 

405.  Cause  of  periodicity. — The   conclusion   can    not  be 
avoided,  that  the  variable  magnitudes  of  stars,  at  least  when 
they  recur  regularly,  are  the  result  of  some  sort  of  revolution. 
More  than  this  is  mere  conjecture.     In  some  cases,  the  star 
may  be  partially  dark  on  one  side,  and  produce  the  changes 
by  rotation  on  its  axis.     In  others,  there  may  be  opaque  bodies, 
either  single  or  existing  in  groups  or  zones,  revolving  about 
the  central  star. 

Newton  suggested  that  the  sudden  appearance  of  a  tem- 
porary star  might  be  the  result  of  a  comet  falling  upon  the 
central  body,  which  was  before  invisible,  and  causing  confla- 
gration. 

406.  Clusters  of  stars. — The  fixed   stars   are   frequently 
grouped  together  in  clusters,  such  as  the  Pleiades,  in  Taurus ; 
Presepe,  in  Cancer ;  and  Coma  Berenices.     If  a  telescope  of 
low  power  is  used,  the  number  of  stars  appears  greatly  in- 
creased.    Figure  1  in  PL  IV.  gives  a  telescopic  view  of  the 
Pleiades. 

There  are  others  which  to  the  naked  eye  appear  nebulous, 
but  by  the  use  of  the  telescope  are  plainly  seen  to  be  clusters ; 
and  in  some  of  them  the  stars  are  so  numerous  as  not  to  be 
easily  counted.  The  clusters  in  Perseus  and  Hercules  are  fine 
examples.  For  the  latter,  see  PL  IY.,  Fig.  3 ;  a  is  its  appear- 
ance with  a  low  power ;  b  is  the  central  part  of  it  with  a  high 
power. 

407.  Nebulas. — These  are  faint  patches  of  light,  having  gen- 
erally an  ill-defined  edge,  and  in  ordinary  telescopes  presenting 
the  same  nebulous  aspect  which  the  closer  clusters  do  to  the 
naked  eye.      As  the  powers  of  the  telescope  are   increased, 
many  nebulae  are  resolved  into  clusters  of  stars,  while  many 
others  retain  their  nebulous  appearance  under  every  power  yet 
employed.     The  number  of  nebulae  now  known  exceeds  4,000. 


208  FOKMS  OF   NEBULA. 

Their  forms  are  exceedingly  various ;  and  in  some  cases  they 
seem  in  this  respect  to  be  greatly  changed  as  the  telescope  is 
improved  in  its  magnifying  and  defining  powers. 

Since  every  advance  which  is  made  in  the  construction  of 
instruments  resolves  some  nebulae  into  clusters  of  stars,  many 
astronomers  have  been  led  to  suppose  that  all  nebulae  are  clus- 
ters, only  too  remote  to  be  resolved  by  means  hitherto  em- 
ployed. Some  facts,  however,  connected  with  this  class  of 
bodies  seem,  to  indicate  that  there  are,  in  some  regions  of  space, 
immense  tracts  occupied  with  nebulous  matter  not  yet  formed 
into  stars. 

4O8.    Varieties  of  form  among  nebulas. — 

1.  Globular.  A  large  number,  especially  of  the  smaller  neb- 
ulae, present  a  circular  outline,  and  grow  brighter  gradually 
from  the  circumference  toward  the  center,  thus  suggesting  the 
idea  of  a  spherical  form.     The  nebulous  stars,  so  called,  differ 
from  them  in  that  the  nebulosity  continues  nearly  uniform  up 
to  a  central  star.     The  planetary  nelidce  have  a  well-defined 
edge,  and  no  bright  center,  and  therefore  bear  some  resem- 
blance to  a  planet. 

2.  Elliptical.  Several  nebulae  present  the  appearance  of  an 
oblate  spheroid  seen  edgewise.     The  most  remarkable  example 
is  the  great  nebula  of  Andromeda.     Its  length  is  1^°,  and  it  is 
easily  seen  by  the  naked    eye  (PI.  IV.,  Fig.  2).      The  dumb- 
bell nebula,  between  Cygnus  and  Aquila,  appears  in  the  best 
telescopes  to  have  an  elliptical  shape.     The  brightest  part  of 
it  has  a  form  slightly  resembling  a  dumb-bell,  or  an  hour-glass. 
(PI.  II.,  Fig.  4). 

3.  Spiral.  This  description  of  nebulae  is  becoming  rather  nu- 
merous since  the  latest  improvements  in  telescopes.      Some 
nebulas  of  very  irregular  shape,  as  formerly  described,  exhibit, 
in  the  best  instruments  of  this  day,  delicate  appendages  having 
a  spiral  arrangement.     The  whirlpool  nebula,  near  the  tail  of 
Ursa  Major,  is  the  most  remarkable  instance  of  this  form  (PL 
IV.,  Fig.  5).     The  crab  nebula,  in  Taurus,  may  yet  be  found  to 
belong  to  this  class  (PL  IV.,  Fig.  4). 

4.  Annular.  A  few  nebulas  have  an  outline  nearly  circular 
or  elliptical  ;  but  appear  more  luminous  on  the  edges  than  in 


MAGNITUDES  OP  NEBULA.  209 

the  central  part.  Such  are  called  annular  nebulae.  The  ap- 
pearance is  that  of  a  hollow  sphere  or  spheroid ;  in  which  case 
we  look  through  the  greatest  depth  near  the  edges.  An  inte- 
resting example  is  situated  in  Lyra,  midway  between  /3  and  y 
(PL  IL,  Fig.  3). 

5.  Irregular.  Besides  the  foregoing  forms,  which  are  all  in- 
dicative of  a  central  force,  and  of  revolution,  there  are  various 
shapes  of  great  irregularity.  None  is  so  celebrated  as  the  great 
nebula  of  Orion,  which  has  been  a  subject  of%  observation 
and  record  for  more  than  two  centuries.  It  becomes  more  ex- 
tended and  more  complex  with  every  new  improvement  in  tel- 
escopes. 

s 

409.  Magnitude  of  clusters  and  nebulce. — Every  cluster  of 
stars,  whether  a  complex  system  of  suns  or  not,  must  occupy 
an  immense  space.     They  are  at  least  as  far  distant  as  the 
nearest  star,  and  how  much  further  we  can  not  know,  and  yet 
they  fill  a  sensible  angle,  and  some  of  them  a  large  one.     It  is 
easy,  therefore,  to  assign  the  lowest  limit  for  their  dimensions. 
The  length  of  the  nebula  in  Andromeda  is  1J°.     Supposing  it 
as  near  as  a  Centauri,  its  absolute  length  must  be  6,000  times 
the  distance  from  the  earth  to  the  sun.     And  if  it  be  many 
times  further  from  us  than  the  nearest  star,  which  is  far  more 
probable,  then  its   dimensions   must   be  just  so  many  times 
greater. 

410.  Changes  in  the  nebulce. — In  repeated  instances  it  has 
been  thought  that  the  forms  of  certain  nebulae  had  essentially 
altered  since  their  discovery.     But  this  is  not  certain  ;  for  it  is 
found  that  the  same  nebula  assumes  a  new  aspect  as  the  tele- 
scope is  improved,  because  some  of  the  more  delicate  features, 
which  were  not  before  noticed,  are  brought  to  view.     It  may 
be,  therefore,  that  all  apparent  changes  of  form  hitherto  noticed 
are  to  be  explained  in  this  way. 

But  there  are  a  few  faint  nebulae,  which  are  known  to  have 
grown  more  dim  within  a  short  time ;  for  they  can  not  now  be 
seen  by  the  same  instruments  which  only  a  few  years  ago 
brought  them  distinctly  into  view.  In  one  or  two  in- 
stances, a  nebula  has  entirely  ceased  to  be  visible.  Such 

14 


210  THE    GALAXY. 

bodies  may,  perhaps,  have  regular  changes,  like  the  periodic 
stars. 

411.  The  galaxy. — This  is  a  belt  or  zone,  of  nebulous  ap- 
pearance, which  encircles  the  heavens,  nearly  coincident  with 
a  great  circle,  and  cuts  the  plane  of  the  equator  at  an  angle  of 
63°.     It  is  usually  called  the  milky-way.     Near  the  constella- 
tion Cygnus,  it  divides  into  two  parts,  which  continue  separate 
nearly  a  semicircle  (150°),  and  then  reunite.     Its  edges  are 
generally  ill-defined,  and  also   quite   crooked   and   irregular, 
having  many  projections  and  indentations. 

The  telescope  shows  that  the  whiteness  of  the  galaxy  is  due 
to  unnumbered  stars,  too  faint  to  be  seen  individually.  Their 
distribution  is  quite  unequal ;  the  stars,  in  some  parts,  being 
crowded  very  closely  together,  while  here  and  there  spaces  oc- 
cur which  contain  but  few.  These  inequalities  are  most  marked 
in  the  southern  hemisphere.  A  small  portion  of  the  southern 
galaxy  is  shown  in  PL  II.  In  the  most  luminous  parts,  Sir 
William  Herschel  estimated  that,  within  an  area  less  than  ^ 
part  of  the  hemisphere,  there  passed  the  field  of  his  telescope 
50,000  stars,  large  enough  to  be  distinctly  seen.  The  whole 
number  of  stars  in  the  milky -way  is  to  be  reckoned  by  millions. 

It  appears,  therefore,  that  by  far  the  largest  part  of  the  stars 
which  are  within  the  reach  of  our  vision  lie  in  a  thin  stratum 
or  ring,  in  the  plane  of  which  the  sun  is  situated.  As  we  our- 
selves, being  near  the  sun,  are  in  this  plane,  we  see  the  stars 
mostly  crowded  into  the  zone  or  belt  which  is  called  the 
galaxy,  while  over  the  other  parts  of  the  sky  they  are  more 
sparsely  distributed. 

412.  The  nebular  hypothesis — What  it  propose*. — The  hy- 
pothesis which  is  known  by  the  name  of  the  nebular  hypothesis 
proposes  to  explain  in  what  manner  the  bodies  composing  the 
solar  system  may  have  arrived  at  their  present  state,  as  to  mo- 
tion, condition,  and  mutual  relations,  through  the  operation  of 
known   laws,   which   the   Creator  has    employed   during  the 
almost  countless  ages  since  the  material  was  at  first  formed. 

413.  Argument  from  analogy* — The  organized  bodies  on 


THE   NEBULAR    HYPOTHESIS.  211 

the  earth,  whether  animal  or  vegetable,  are  not  created  in  their 
mature  and  perfect  state,  performing  at  once  all  the  functions 
for  which  they  were  designed ;  but  they  groiv  to  this  condition 
by  a  series  of  changes,  which  extend  generally  through  a  num- 
ber of  years. 

So  the  soils  of  the  earth  were  not  first  formed  in  their  present 
condition,  fitted  to  sustain  the  vegetation  which  clothes  them ; 
but  are  the  result  of  slow  disintegration  of  the  rocky  mountain 
tops,  through  the  action  of  water  and  changes  of  temperature. 

It  is  more  in  accordance  with  the  Creator's  plan  of  operation, 
so  far  as  we  can  discover  it,  that  the  sun,  planets,  and  satellites 
should  have  been  brought  into  their  present  condition  through 
a  long-continued  course  of  change,  than  that  they  should  have 
been  created  and  set  in  motion  as  we  now  see  them. 

414.  facts  in  the  solar  system  which  form  the  basis  of  the 
hypothesis. — 

1.  The  sun,  the  planets,  and  the  satellites,  so  far  as  they  are 
known  to  rotate  at  all  on  their  axes,  rotate  nearly  in  the  same 
direction,  from  west  to  east.    And  the  revolutions  of  all  planets 
about  the  sum,  and  of  all  satellites  about  their  primaries,  with 
but  few  and  trifling  exceptions,  revolve  in  the  same  general  di- 
rection, from  west  to  east. 

2.  The  sun,  which  contains  nearly  the  whole  material  of  the 
system,  is  a  sphere  in  a  condition  of  intense  heat.    The  interior 
of  the  earth  is  in  a  red-hot  melted  state,  as  is  proved  by  the 
volcanoes  on  its  surface.     The  moon  is  covered  with  volcanic 
craters,  which  show  that  it  is,  or  has  been,  in  the  same  condi- 
tion, internally,  as  the  earth  now  is. 

415.  The  nebular  hypothesis  stated. — It  assumes  that  the 
whole  space  occupied  by  the  solar  system,  and  extending  far 
beyond  its  present  limits,  was  filled  with  nebulous  matter,  in  an 
exceedingly  rare  and  intensely  heated  condition  ;  and  that  this 
entire  mass  was  put  into  a  state  of  rotation  in  the  direction 
which  we  now  call  from  west  to  east. 

This  assumption  being  made,  the  following  consequences 
would  ensue,  during  the  lapse  of  immense  periods  of  time,  in 
accordance  with  the  well-known  laws  of  the  material  creation. 


212  THE   NEBULAR    HYPOTHESIS. 

By  gravity  and  the  centrifugal  force,  the  vast  nebula  takes  a 
spheroidal  shape. 

Heat  is  radiated  from  its  exterior  into  the  boundless  space 
around  it ;  and  by  this  loss,  the  nebula  contracts  in  diameter. 
But  as  it  contracts,  the  given  velocity  of  rotation  at  the  surface 
causes  a  quicker  rate  of  revolution,  until,  at  length,  the  cen- 
trifugal force  of  the  equatorial  part  equals  the  attraction  toward 
the  center  of  the  entire  mass.  As  soon  as  these  two  forces  are 
equal,  the  equatorial  part  rotates  independently  of  the  interior, 
while  the  latter  contracts  still  further,  and  leaves  the  superfi- 
cial part  revolving  as  a  nebulous  ring. 

After  the  central  portion  has  left  the  ring,  it  goes  on  con- 
tracting as  before,  till  it  leaves  a  second  ring.  Thus,  an  indefi- 
nite number  of  concentric  nebulous  rings  may  be  left,  each 
revolving  from  west  to  east,  and  at  a  swifter  rate  according  as 
it  is  nearer  the  center.  The  central  mass,  which  thus  succes- 
sively deposits  its  rings,  is  the  sun  of  the  system. 

416.  "While  the  material  composing  each  ring  goes  on  cool- 
ing and  contracting,  unless  the  quantity  is  exactly  equal  on 
every  side,  which  is  improbable,  the  whole  of  it,  at  length,  is 
drawn  toward  the  heaviest  side,  until  it  is  gathered  into  a 
spheroid,  revolving  once  on  its  own  axis,  while  it  revolves  once 
around  the  central  mass.  These  spheroids  are  the  planets,  re- 
volving around  the  sun. 

But  as  the  planetary  spheroid  continues  to  contract  by  cool- 
ing, its  rate  of  rotation  is  quickened,  until  it  leaves  its  equa- 
torial part  revolving  in  a  ring  about  it,  in  the  same  manner  as 
the  central  nebula  has  done ;  and  this  it  may  do  in  repeated 
instances. 

These  subordinate  rings  are  likely  also  to  collect  into  so 
many  spheroids,  revolving  about  the  larger  ones,  and  on  their 
own  axes.  These  are  satellites.  In  case  the  parts  of  a  ring  are 
very  exactly  balanced,  they  may  preserve  their  condition  of  a 
ring,  instead  of  gathering  into  a  satellite.  An  example  is  seen 
in  the  ring  of  Saturn. 

It  is  conceivable  that  a  multitude  of  small  rings,  instead  of 
one  large  one,  may  be  detached  from  the  central  mass  when 
the  separation  occurs.  This  seems  to  have  been  the  case  in 


THE  NEBULAR  HYPOTHESIS.  213 

the  formation  of  those  rings  from  which  the  planetoids  were 
formed. 

417.  After  the  planets  and  satellites  have  cooled  sufficiently, 
they  become  non-luminous  bodies,  and  are  gradually  changed 
from    nebulous  into  a  liquid  or  solid  condition.     And,  in  a 
given  case,  the  exterior  may  be  solid,  while  the  interior  re- 
mains in  a  liquid  and  highly  heated  condition.     This  is  the 
present  state  of  the  earth,  and  the  present  or  recent  condition 
of  the  moon. 

That  the  planes  of  motion  throughout  the  system  are  not  co- 
incident, is  to  be  ascribed  to  disturbing  influences  which  the 
several  bodies  have  been  exerting  on  each  other  during  the 
vast  periods  of  time  that  have  elapsed  since  they  were  detached 
from  the  solar  mass. 

418.  Application  to  other  systems. — Every  fixed  star  which 
is  single  may  be  the  condensed  nucleus  resulting  from  an  op- 
eration similar  to  that  which  has  been  described;   and    the 
double  and  triple  stars  may  be  considered  as  cases  in  which 
either  the  nebula  became  divided  into  two  or  three  parts,  be- 
fore the  contraction  had  proceeded  far,  or  else  the  nebulous 
mass,  being  very  oblate,  a  large  part  of  it  was  detached  at 
once,  and  collected  into  a  planet,  nearly  equal  to  the  central 
part. 

The  nebulae  of  regular  form,  not  capable  of  being  resolved 
into  separate  stars,  may  still  be  in  the  condition  of  the  solar 
system  before  its  rings  began  to  be  separated  from  the  original 
body. 


214 


TABLE   L— THE   CALENDAR. 


A  TABLE  TO  FIND  THE  DAY  OF  THE  WEEK  OF  ANY  GIVEN  DATE,  FROM  THE  YEAB  5000  B.  c. 
TO  THE  YEAR  2700  OF  THE  CHRISTIAN  ERA. 


CENTUKIKS  BEFORE  CHRIST. 

NEW  { 
STYLE  1 

OLD    I 
STYLE  j 
I 

CENTURIES  AFTER  CHRIST 

4SOO 
4100 
3400 
2700 
2000 

i:$oo 

600 

4700 
4000 
3300 
2600 
1900 
1200 
500 

~D~ 

4000 
3900 
3200 
2500 
1800 
1100 
400 

4500 
3800 
3100 
2400 
1700 
1000 
300 

4400 
3700 
3000 
2300 
1600 
900 
200 

5000 
4300 
3600 
2900 
2200 
1500 
800 
100 

4900 
4200 
3500 
2800 
2100 
1400 
700 
0 

1700 
2100 

1800 
2200 

300- 
1000- 
1700- 
2400' 

1500 
1900 
2300 

1600. 
2000- 
2400' 

0- 
700- 
1400- 
WOO- 

100- 
800- 
1500- 
2200« 

200- 
900' 
1600- 
2300- 

400' 
1100« 
1800' 
2500' 

BOO- 

1200' 
1900- 
2600' 

600- 
1300« 
2000- 
2700. 

C 

E 

F 

G 

A 
B 

B 

0 

28' 

56* 

•57 
~58~ 

84- 
•85 
~SQ~ 

c 

~B~ 

D 

E 

F 

G 

A 

B 

D 

E 

F 

G 
B 
~C~ 

A 

C 
E 

•1 

~2~ 

•29 
~30~ 

C 

^B~ 

D 
C 

E 

~rT 

F 

G 
F 

A 

F 
G 
A 

G 

~T 

A 
~B~ 

C 
~D~ 

D 

A 

E 

G 
F 

E 

F 

3 

31 

32^ 

59 
~60^~ 

87 

G 
E 

A 

B 

c 

D 

E 

B 

C 

D 

E 
~G~ 

E 

F 

G 

G 
A 
~C~ 

4? 

•5 

88- 

F 

G 
F 

A 

~cT 

B 

C 

D 

B 

C 

D 

F 
A 

•33 

•61 

•89 
"90~ 

D 

E 
D 

A 

B 
~A~ 

C 

D 
E 

E 

~F~ 

F 
~G~ 

B 

6 

7 

34 
35 

62 
~63~ 

C 
~B~ 

E 

F 

G 

"F^ 

B 

A 
~F~ 

A 

B 

C 
D 

D 

91 
^2^ 
^93~ 

C 

D 
B 

E 

~c~ 

G 

F 

G 

A 

B 
~C~ 

~&~ 

C 

E 
F 

8- 

36- 

64- 
•65 

G 

A 
G 

D 

E 

G 

A 
C 

B 
D 
^E~ 

D 

F 

E 

•9 

~uT 

•37 

F 

~E~ 

A 

B 
A 

C 

D 
C 

E 

B 

~cT 

G 

A 

38 

66 

94 

F 

G 

B 

D 

D 

F 
G 
A 

G 

A 

B 
C 
~D~ 

11 

39 

67 

~68^~ 

95 

D 
B 

E 
C 
~B~ 

F 

G 

A 

B 

C 
~t^ 

D 

E 

F 

A 

~1T 

B 
~C~ 

12- 

40- 

^4T 

96- 

D 
C 

E 

F 
E 
D 
~C~ 

G 
F 

E 
G 
A 

F 
A 

G 

•13 

•69 

•97 

A 

D 
~C~ 

G 

B 
~C~ 

C 

D 

E 

F 

14 

42 

70 

98 
~99~ 

G 

A 

B 

E 
~D~ 

F 
E 
~C~ 
~B~ 

B 

D 

E 

F 

G 
A 
^B~ 

15 

43 

44. 

^45~ 

71 

~72^~ 

F 
D 

G 

A 
F 

B 

B 

C 

D 

E 
^r 

F 
G 

G 
~A~ 

16* 
^17 

E 
D 

G 

A 

B 

C 

E 

D 
^F~ 

E 
~G~ 

•73 



C 

E 

F 
~E~ 
^D~ 

G 

A 
~G~ 

A 

B 

C 

D 

18 

46 

74 
~75~ 

B 

C 

D 

F 

E 

A 

F 

G 

A 
B 

B 

C 

D 
E 

E 
^F~ 

19 

47 
~48^ 

A 
^F~ 

B 

C 
A 

F 

G 

G 

A 

C 

D 

20* 

76- 

G 

B 

C 

D 

E 

A 
~C~ 

B 
D 

C 

D 

E 

F 

G 

•21 

•49 

•77 



E 

F 
E 

G 

A 

B 
A 
~G~ 

C 
B 
~A~ 

D 

E 

F 

G 

A 

B 

22 

50 

78 
79 

D 

F 
E 

~cT 

G 

C 
~1T 
~G 

D 

E 

F 

G 

A 
~B~ 

B 

C 
~^D~ 

C 
D 

23 

51 
^2^ 

C 

D 

F 

E 

F 

G 
A 
~C~ 

A 
^B~ 

24- 
^25~ 

80- 

A 
~G~ 

B 
A 
G~ 

D 
C 

^iT 

^T 

E 

F 

F 
A 
B 

G 

C 

E 

•53 

•81 

B 

D 

E 

F 

B 

C 

D 

E 

F 

G 

26 

54 

82 

F 

A 

C 

D 

E 
D 

D 

E 

F 

G 

A 

27 

55 

83 

E 

F 

G 

B 

C 

TABLE   I. — THE   CALENDAR. 


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216 


ELEMENTS  OF  THE  PLANETS. 


TABLE  II.    ELEMENTS  OF  THE  PLANETS. 

(Principally  from  Le  Verrier  and  Herschel.) 


NAME. 

1 

& 

Relative 
Distances. 

Eccen- 
tricity. 

Sidereal 
Revolution 
in  days. 

Synodical 
Revolution 
in  days. 

Mean  daily 
motion. 

Inclination 
of 
Orbit 

Mercury.  . 

5 
2 
5 
$ 
U 
% 
W 
t 

0.387099 
0.723332 
1.000000 
1.523691 
5.202798 
9.538852 
19.182639 
30.036970 

.205618 

.006833 
.016770 
.093262 
.048239 
.055996 
.046577 
.008719 

87.969 
224.701 
365.256 
686.980 
4332.585 
10759.220 
30686.821 
60126.720 

115.877 
583.921 

779.936 
398.884 
378.092 
369.656 
367.489 

4°  5'  32".6 
1  36    7  .7 
0  59    8  .3 
0  31  26  .5 
0    459  .1 
0    2    0  .5 
0    0  42  .2 
0    021  .6 

7°  0'    8" 
3  23  31 
000 
1  51     5 
1  18  40 
2  29  28 
0  46  30 
1  46  59 

Venus 

Earth  .  . 

Mars  

Jupiter  
Saturn  

Uranus 

Neptune  .  . 

NAME. 

Rotation 
in 
hours. 

Mean 
Diameter 
in  miles. 

Mean 
angular 
Diameter. 

Relative 
Volume. 

Relative 
Mass. 

Relative 
Density. 

Relative 
Gravity. 

Sun 

610 

886  000 

1922"  5 

1405  000 

354936 

0  25 

OQ  Q1 

Mercury  . 

24.91 

3,100 

8. 

0.060 

01183 

1.97 

0.77 

Venus  .  .  . 

23.55 

7,800 

17. 

0.958 

0.8832 

0.92 

0.91 

Earth  .  .  . 

24.00 

7,912 

1.000 

1.0000 

1.00 

1.00 

Mars  

24.66 

4,500 

6. 

0.184 

0.1324 

0.72 

0.41 

Jupiter  .  . 

9.92 

91,000 

37. 

1521. 

338.0342 

0.22 

2.55 

Saturn  .  . 

10.48 

77,000- 

16. 

921. 

101.0637 

0.11 

1.07 

Uranus  .  . 

35,000 

4. 

87. 

14.7889 

0.17 

0.76 

Neptune  . 

34,000 

2. 

79. 

24.6483 

0.31 

1.33 

TABLE  III.    DISTANCES  AND  PERIODS  OF  THE  PLANETOIDS. 

(From  Sup.  Berlin  Astr.  Jahrbuch,  1867,  and  Astr.  Nachr.,  1865.) 


No. 

NAME. 

Relative 
Dis- 
tance. 

Sidereal 
Period 
in  days. 

No. 

NAME. 

Relative 
Dis- 
tance. 

Sidereal 
Period 
in  days. 

1 

Ceres  

2.7663 

1680.96 

2 

Pallas 

27696 

1683  51 

S 

Juno  

2.6707 

1594.20 

4 

Vesta 

23609 

1325  03 

5 

Astrsea  

25772 

1511  18 

6 

Hebe 

2  4244 

1378  85 

7 

Iris  

23862 

1346  36 

g 

Flora 

2  2014 

1  1QR  01 

9 

Metis 

23866 

1346  72 

10 

Q  1R10 

OAJQ  1f» 

11 

Parthenope  .  .  . 

2.4519 

1402  36 

12 

Victoria 

2  3342 

1302  73 

18 

Egeria  

25775 

1511  45 

14 

Irene 

2  5860 

ifjiG  07 

15 

Eunomia  

2.6436 

1570  04 

16 

Psyche 

2  9233 

1825  97 

17 

Thetis  

24737 

1421  07 

18 

o  9Q5fi 

1  270  44 

19 

Fortuna  

2.4416 

1393.48 

20 

Massilia  ...... 

24144 

1365  53 

21 

Lutetia  

24411 

1388  24 

22 

o  QOQ2 

1812  41 

23 

Thalia  

26280 

1555  28 

24 

Themis 

3  1431 

2035  29 

?,5 

Phocea     

24008 

1358  74 

26 

0  fiKfil 

1  ^81  OQ 

PLANETOIDS. 


TABLE  III.  (continued).    PLANETOIDS. 


No. 

NAME. 

Relative 
Dis- 
tance. 

Sidereal 
Period 
in  days. 

No. 

NAME. 

Relative 
Dis- 
tance. 

Sidereal 
Period 
in  days. 

97 

Euterpe  .    .  . 

2.3468 

1313.17 

98 

Bellona  

2.7785 

1691  63 

29 
81 

Amphitrite.  .  .  . 
Euphrosyne 

2.5544 
31513 

1491.22 
2044  64 

30 
39 

Urania  
Pomona  

2.3655 

25868 

1328.85 
151964 

38 

Polyhymnia  .  . 

2.8653 

1770.62 

34 

Circe  

2.6865 

1608.34 

85 

Leucothea 

30040 

1904  23 

36 

Atalanta 

27458 

1661  88 

87 

Fides 

2.6414 

1567  97 

38 

Leda       

27401 

1656  76 

89 

Laetitia  

2.7740 

1687.62 

40 

Harmonia  

2.2677 

1247.33 

41 

Daphne 

2.7657 

1679  93 

42 

Isis  .    .  . 

24400 

1392  17 

48 

Ariadne  

2.2034 

1194  65 

44 

Nysa  

24234 

1377  98 

45 

Eugenia 

27218 

1640  14 

46 

Jjestia 

25262 

1466  52 

47 

A°"laia 

2.8812 

1786  36 

48 

Doris  .... 

3  1094 

2002  69 

49 

Pales  

3.0825 

1976  60 

50 

Virginia  

26491 

1574  87 

51 

Nemansa  

2.3657 

1329.03 

59 

Europa  

3.1000 

1993.58 

58 

Calypso 

2.6188 

1547  95 

54 

Alexandra 

27123 

1631  58 

55 

Pandora  

2.9591 

1673  95 

56 

Melete  . 

25959 

1527  67 

57 

59 

Mnemosyne  .  .  . 
Olympia 

3.1565 

27131 

2048.40 
1632  30 

58 
60 

Concordia  
Echo. 

2.7014 
23931 

1621.76 
1352  18 

61 

Danae  

2.9837 

1882  45 

69 

Erato  

31297 

2022.30 

68 

Ausoiiia 

28949 

1353  80 

64 

Angelina 

26805 

1602  97 

65 

Maximiliana.  . 

3.4205 

2310  65 

66 

Maia 

26635 

1587  77 

67 

Asia  

2.4217 

1376  54 

68 

Leto  

2.7836 

1696  31 

69 

Hesperia  . 

29707 

1871  12 

70 

Panopea 

26129 

1543  03 

71 

Niobe  

2.7501 

1665  83 

79 

Feronia  

22654 

124541 

78 

Clytie 

26666 

1590  49 

74 

Galatea 

27777 

1690  93 

75 

Eurydice  

2.6707 

1594  19 

76 

Freia 

33864 

2276  20 

77 

Friffsra  

2.6719 

1595  27 

78 

Diana  

26236 

1552  23 

79 

Eurynome  .    .  4 

24437 

1395  29 

80 

Sappho 

22971 

1271  64 

81 

Terpsichore  .  .  . 

28591 

1765  75 

82 

Alcmene  

27547 

1669  98 

88 

Beatrice     .... 

24287 

1382  52 

84 

Clio 

23674 

1330  52 

85 

2^6465 

1584.03 

218 


ELEMENTS   OF  THE   SATELLITES. 


TABLE  IV.    ELEMENTS  OF  THE  SATELLITES 

(Principally  from  Herachel.) 


THE  MOON. 

Mean  distance  : 
Mean  sidereal  r 
Mean  synodical 
Mean  revolutioi 
Mean  revolutioi 
Mean  inclinatio 
Eccentricity  of 
Mean  diameter 
Diameter,  (eartl 
Surface,  (earth's 
Volume,  (earth'i 
Density,  (earth's 
Mass,  (earth's  = 

rom 
evolu 
revo 
lof  n 
i  of  a 
tiof  < 
orbit 
of  th 
I's  = 
=  1 

3  =  1 
3  =  1 

-  !"> 

the  earth,  (miles)  

238650 
J7.32166 
J9.53059 
)3.39108 
52.57534 
0  8'  48" 
.054908 
2160 
0.2730 
0.0745 
0.0203 
0.5565 
0.0116 

tion,  (days)   i 

lution,  (days)  i 

odes,  (days)  671 

psides,  (days)  32J 

)rbit  to  ecliptic  5 

e  moon,  (miles)  

1)  

)  .  .                                                      ,    l  or 

)  ..a'TrOr 

)  

J-     c\r 

"  -f-                                                                                                   '  '  86    "* 

1 

a 
& 

Sidereal 
Revolution. 

Distance 
in  radii 
of  Planet 

Distance 
in 

miles. 

Diameter 
in 

miles. 

Satellites 
of 
Jupiter. 

1 
2 
3 

4 

d       h      m         s 
1      18     27     34 

3    13    14    36 
7      3    42    33 
16    16    31    50 

6.04853 
9.62347 
15.35024 
26.99835 

275000 
438000 
698000 
1229000 

2440 
2190 
3580 
3060 

Satellites 
of 
Saturn. 

1 
2 
3 
4 
5 
6 
7 
8 

0    22    37    23 
1      8    53      7 
1    21    18    26 
2    17    41      9 
4    12    25    11 
15    22    41    25 
21      7      7    41 
79      7    53    40 

3.3607 
4.3125 
5.3396 
6.8398 
9.5528 
22.1450 
26.7834 
64.3590 

129000 
166000 
206000 
263000 
368000 
853000 
1031000 
2478000 

1200 
3000 

1800 

Satellites 
of 
Uranus. 

1 
2 
3 

4 

2    12    29    21 
4      3    28      8 
8    16    56    31 
13    11      7    13 

7.40 
10.31 
16.92 
22.56 

130000 
181000 
297000 
396000 

Satellite  of 
Neptune. 

5    21      2    43 

12. 

204000 

PLATE  II. 

COMET   OF  1858.-NEBULjE. 


rLATK  111. 

PART  OF   GALAXY.— DOUBLK  STAIIS 


1.  Castor.         2.  y  Leonis.      3.  31)  Drac.     4.  A  Opli.     5.  11  Mon  >c.      0.  $  Caucri. 


Revolutions  of  /  Virginis. 


1837.        1838.          1839.        1840.         1845.          1850.        1860.        Orbit. 


PLATE  IV. 

CLl  STKRS.- NEBULA. 


v£v-  -*Vi 


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